Differential Calculus From Scratch

Learn to evaluate limits of rational functions by simplifying with factoring, canceling common factors (like x-3 from x^2-9), and substituting the simplified form.

Show how to evaluate limits with surds by rationalizing with the conjugate, applying the difference of squares to cancel terms, and substituting to resolve 0/0 indeterminate forms in multiple examples.

Apply conjugate multiplication to resolve 0/0 limits in differential calculus, then substitute to obtain: as x approaches 2, the limit is 1/2; as x approaches a, the limit is 2/(3√3).

Explore five fundamental limit theorems: sum and difference of limits, constant multiples, product of limits, and quotient of limits, to solve limit problems.

Explore essential series expansions, such as (1+x)^n, e^x, log(1+x), sin x, and cos x, and apply them to simplify and solve limits.

this lecture demonstrates how to solve limits with exponential expansions, applying the e^x and e^{-x} series to compute limits such as (e^x−1)/x→1 and (e^x+e^{-x}−2)/x^2→1, including other x→0 limits.

Apply series expansions to evaluate limits: binomial expansion of (1+x)^n, log(1+x) and a^x, cancel common x terms, and obtain limits such as (1+x)^n-1 over x and log(x)/(x-1).

Demonstrate the theorem that the limit of (x^n − a^n)/(x − a) as x tends to a equals n a^{n−1}, by expanding, substituting x = a, and summing terms.

Use the limit theorem (x^n - a^n)/(x - a) → n a^(n-1) to evaluate limits for various a and n, with examples like -2 and 3.

The lecture proves the limit theorems sin x over x tends to 1 and tan x over x tends to 1, then applies them to sin(2x)/x and sin(5x)/(3x).

Explore solving limits from scratch in calculus using 1 - cos x = 2 sin^2(x/2) to evaluate as x tends to zero, including (1 - cos x)/x.

Using limits and trig identities, the solution simplifies the expression and shows the limit as theta tends to zero equals m squared over n squared.

This lecture demonstrates evaluating limits as x approaches zero for expressions involving tan x, sin x, and cos x, using identities and standard limits to obtain values such as 2 and 1/2.

Explore solving limits as x approaches zero in differential calculus, using conjugate multiplication and identities like one minus cos x equals sin^2 x, to simplify expressions with sine and cosine.

Evaluate the limit as x tends to zero by applying trig identities, such as 1 minus cos 2x equals twice sine squared x, yielding minus one.

practice evaluating limits as x approaches zero using trig identities such as sin x / x equals 1 and cos x tends to 1, with division by x and substitution.

Explore four key trigonometric formulas (sum and difference of sines and cosines) and apply them to evaluate limits of trig functions.

Demonstrate solving multiple trig limits from scratch by applying sum-to-product identities and standard limits to derive explicit results.

Develop mastery of two core trigonometric limit results and apply them to evaluate limits, using substitutions like sin(pi minus x) and cos x transformations.

Compute limits of trigonometric expressions at key angles using algebraic manipulation and standard limits, covering sine, cosine, and tangent near pi by four and five by six.

Learn to evaluate limits at infinity by substituting x=1/y with y tending to 0, apply highest-power method for rational functions, seeing 1/x^2 tends to 0 and (x-1)/x tends to 1.

Demonstrates evaluating diverse limits from infinity to zero using binomial expansion, conjugates, and substitutions, including e^a, sine and cosine limits, and asymptotic rational functions.

Explore three fundamental limit formulas, including log base e results, and apply them to solve varied limits using algebraic tricks, conjugates, and small-angle sine expressions.

Evaluate the limit as x tends to one by dividing by x-1 and using sine identities, yielding minus one over pi times log A base e.

Examine the existence of limits by comparing left and right hand limits, and solve examples like |x-4|/(x-4) and (x - |x|)/x to determine limit existence.

Explore how to determine the limit of a piecewise function at x = 1 by comparing the left-hand and right-hand limits, showing they differ and the limit does not exist.

Demonstrate the limit as x tends to 1 exists by equating left and right limits to 1 with piecewise definitions, then deduce A=0 and B=4.

Explain how to test continuity at a by ensuring the limit as x approaches a exists and equals f(a), and identify breaks or jumps.

Check continuity by matching left and right limits to f(2). x^3 is continuous at 2 with limit 8; the cases at 4 and 0 are not continuous.

Show that the greatest integer function is not continuous at x = n by evaluating left and right limits: floor(n - h) tends to n - 1 and floor(n + h) tends to n.

Show how the continuity of a piecewise function at zero is established by comparing left and right limits, both equal to two, with f(0)=2.

Explains open and closed intervals and how a function is continuous on them. Defines endpoint conditions, left-hand limits, and the properties of continuous and everywhere continuous functions.

Investigate the continuity of piecewise functions by evaluating left and right limits at x=0, using polynomial and cos functions, and determine parameter values to ensure continuity.

Determine differentiability of a real-valued function on an open interval at c by the finite limit of (f(x)-f(c))/(x-c). Ensure left-hand and right-hand limits exist and are equal to conclude f'(c).

Differentiability implies a unique tangent at a point, and left and right derivatives must match. For f(x)=x^2, the left and right derivatives at x=1 equal 2, giving derivative 2.

Examine the function f(x) = |x-2| to prove continuity at x=2 and show it is not differentiable by evaluating left and right derivatives, which yield -1 and 1.

Examine why the greatest integer function is not differentiable at x = 1 by comparing left-hand and right-hand derivatives, using limits to show the divergent behavior.

Demonstrates how to prove a piecewise function is not differentiable at x=2 by computing left-hand and right-hand derivatives; left-hand derivative equals 1, right-hand derivative equals -1.

Show left- and right-hand derivatives at x = 0; left derivative is 0 and right derivative is 1, so f is not differentiable at 0.

solve for a linear function f(x) = m x + 1 using f(0) = 1 and f′(0) = 1; apply the limit definition of the derivative to show m = 1, then find f(2) = 3.

Differentiate from first principles by defining the derivative as a limit. It is the slope of the tangent and measures the rate of change of y with respect to x.

Explore differentiation from first principles with two solved examples: differentiate x^3 minus 27 to obtain 3x^2, and differentiate (x minus 1)(x minus 2) to obtain 2x minus 3.

Learn differentiation from first principles by solving derivative limits for functions like 1/x^2, root x, 1/x, and sin 2x, using conjugates and limit evaluation.

Solve differentiation from first principles for several functions: f(x)=sin(x^2) with derivative 2x cos(x^2); f(x)=sin^2 x with derivative sin(2x); and f(x)=x^{-3/2} with derivative -3/2 x^{-5/2}.

Apply first principles to derive the derivatives of f(x)=x sin x and f(x)=e^{x^2}, showing sin x + x cos x and 2x e^{x^2}.

Derive derivatives of common functions from first principles: d/dx x^n = n x^{n-1}, d/dx e^x = e^x, and d/dx log base e x = 1/x.

Derive the derivatives of sin x from first principles using limit definitions and trig identities, then apply the same approach to cos x to show d/dx sine x equals cos x and d/dx cos x equals -sin x.

From first principles, the lecture derives the derivative of tan x by evaluating the limit of [tan(x+h)-tan(x)]/h using sin x and cos x, yielding sec^2 x.

Derive cot x from first principles by setting f(x) = cot x and applying the limit definition, rewriting cot x as cos x over sin x, and simplifying with trigonometric identities.

Show how to obtain derivatives from first principles, including the derivative of sine x and cosine x, then apply these to derive the derivatives of sec x and cosec x.

Learn core derivative rules for power functions, exponentials, logs, and trig functions, and apply them to derivatives like 8x^3, 6√x, and 2^x.

Practice derivative rules through solved examples, differentiating sums of functions like x^3, e^x, 3^x, x^2, and log x, with trig terms such as cos x and sin x.

Apply the product rule to differentiate products of two or more functions, using f′(x)g(x) + f(x)g′(x). Practice differentiating x e^x and x^2 e^x.

Differentiate the function x^2 tan x minus x log x with base e, and extend to e^x sin x and x^p cos x, highlighting derivative techniques.

Master derivative techniques with solved examples, including derivatives of e^x, log x, product rules, chain rule, and roots, across expressions like x(1+log x) and x sine x.

A solved example demonstrates differentiating a function with x and trigonometric terms, applying derivatives of x, cos x, and cosec x, then factoring to obtain a simplified derivative.

Solve and differentiate the function formed by the product of e^x cos x and x^-3, applying the product rule and chain rule to obtain the derivative step by step.

Learn the quotient rule by differentiating f(x)/g(x) as (g(x) f'(x) - f(x) g'(x)) / (g(x))^2, with worked examples like e^x/x and (2x+3)/(x^2-5).

Find the derivative of a function involving square roots, apply derivatives of radical terms, and simplify by using a common denominator and cancellation.

Derive derivatives using quotient, product, and chain rules for functions like cos x over log x, e^x(x−1)/(x+1), (x−1)/(x+1), and sin x/(1−x).

This lecture demonstrates differentiating composite and rational expressions, using chain rule on terms like 1±10x and sine/cosine, ending with -2/(sin x - cos x)^2.

Master the chain rule to differentiate a function of a function by multiplying inner and outer derivatives. Apply to examples such as sin(x^3), e^{sin x}, and 1/√(a^2−x^2).

Explore chain rule applications to differentiate composite functions such as (ax+b)^m, (3x+5)^6, sqrt(ax^2+bx+c), and log(tan(x/2)).

The lecture applies the chain rule to two problems, expressing y as nested functions and using derivatives like 1/(2 sqrt u) and minus cos^2 x to find dy/dx.

Learn to differentiate complex expressions with the chain rule, solving examples like the derivative of sqrt((1-10x)/(1+10x)), sin(sqrt(sin x+cos x)), and one over log cos x.

Apply product and chain rules to differentiate sin(2x) cos(3x), deriving sin(2x) and cos(3x); then differentiate (sin x + x^2)/(2x) by treating it as a product with tan(2x).

Master solved examples of differentiating products and nested functions in differential calculus. Learn product rule with (balance plus log x) and cos(3x), and chain rule for sin(sqrt x).

Differentiate a product using product and chain rules: e^x log(sin 2x) yields e^x[log(sin 2x) + 2 cot(2x)], and e^{ax} cos(bx+C) yields a e^{ax} cos(bx+C) - b e^{ax} sin(bx+C).

Differentiate y = log(log(log x)) using the chain rule with intermediate substitutions, then simplify to a final expression in terms of x, log x, and log log x.

Apply the chain rule to differentiate y = sqrt(log(sin(x^3/3 - 1))). y' = x^2 cos(x^3/3 - 1) / (2 sqrt(log(sin(x^3/3 - 1))) sin(x^3/3 - 1)).

explain the derivatives of inverse trigonometric functions and present essential formulas for sin⁻¹x, cos⁻¹x, tan⁻¹x, and cot⁻¹x, then apply them to differentiate composite expressions.

Apply chain rule to differentiate composite expressions involving inverse trigonometric functions, such as arctan x and sin(arctan x), with step-by-step derivative rules.

Differentiate arccos((1 - x)/(1 + x)) in differential calculus using the chain rule and quotient rule. Obtain the derivative 1/(sqrt{x}(1+x)).

Explore differential calculus from scratch by differentiating inverse functions, including sine inverse x/(x+1), tan inverse, exponential, and logarithmic inverses, using the chain rule.

Learn to differentiate log(sine inverse x) raised to the fourth using the chain rule, with variables u and v and a stepwise dy/dx derivation.

Explore essential trigonometric formulas, including double-angle, triple-angle, and sum formulas, and apply them to evaluate the derivatives of inverse trigonometric functions.

Differentiate inverse trigonometric expressions by applying half-angle simplifications. The lecture demonstrates derivatives such as 1/2 and -1 for functions like tan inverse((1−cos x)/sin x) and related forms.

Learn to differentiate inverse trigonometric functions by simplifying with identities, demonstrating that derivatives of arctan and arccos expressions reduce to one half.

Solve two differentiation problems by simplifying inverse trigonometric expressions. The lecture demonstrates using half-angle identities to rewrite tan inverse forms and compute the derivatives, yielding 1/2 and -1/2.

learn to differentiate inverse trigonometric functions using substitution to simplify, deriving the derivative of arctan x and related identities such as arcsin(2x/(1+x^2)) = 2 arctan x.

Learn to differentiate inverse functions and inverse trig expressions using substitutions like x=tan θ or x=cos θ, deriving arctan, arccos, arccot, and arcsin derivatives.

Solve and differentiate inverse trigonometric functions using substitution to simplify expressions, yielding derivatives such as dy/dx = 1/(2(1+x^2)) and dy/dx = -1/(2(1+x^2)).

Substitute x = tan theta and apply identities to rewrite y as pi/2 - theta/2 with theta = arctan x, then differentiate to obtain dy/dx = -1/(2(1+x^2)).

Solve two differentiation examples by simplifying with x = cos theta, applying cot inverse and sine inverse identities, and obtaining dy/dx = -1/(2(1+x^2)) and -1/(2√(1−x^2)).

Learn to differentiate implicit functions by differentiating terms of x and y and multiplying by dy/dx, then solve for dy/dx in equations like x^2+y^2=9 and root relations.

Apply implicit differentiation to the equation sin^2 x + 2 cos y + x y = 0, differentiate terms, and solve for dy/dx using product and chain rules.

Use trig substitutions to solve implicit equations involving sqrt(1−x^2) and sqrt(1−y^2), and differentiate to obtain dy/dx expressions. Explore two examples yielding dy/dx = sqrt(1−y^2)/sqrt(1−x^2) and dy/dx = −sqrt(1−y^2)/sqrt(1−x^2).

Learn to find dy/dx as the inverse of dx/dy using implicit differentiation, illustrated with a trig example and the equation cos x + y = y sin x.

The lecture demonstrates differentiating an implicit equation with respect to x, using the chain rule on sin(x y) and t = x y, and solving for dy/dx with algebraic simplifications.

Discover how to differentiate y = x^x using logarithmic differentiation, obtaining dy/dx = x^x(1 + log x) by differentiating log y = x log x.

Use logarithmic differentiation to differentiate (sin x)^x and x^{arcsin x}. Extract dy/dx from log y forms log y = x log sin x and log y = arcsin x log x.

Apply logarithmic differentiation to differentiate y = sin x log x and y = sin x raised to x plus cos x raised to x, using product and chain rules.

Master differential calculus from scratch by differentiating y equals x squared tan inverse x over sqrt(1+x^2) using logarithmic differentiation. Compute dy/dx through the logarithmic derivative.

Apply logarithmic differentiation to the relation y log x = x log y and solve for dy/dx, yielding dy/dx = y/x.

Learn to differentiate infinite power towers using logarithmic differentiation. Derive dy/dx for y = x^(x^(...)) and related expressions, obtaining dy/dx = y^2/x (1 − y log x).

Explore differentiation of parametric functions by computing dy/dx as (dy/dt)/(dx/dt), using dt/dx when needed, with x = a + sin t and y = a(1 - cos t).

Practice solving differential calculus problems by applying the chain rule to parametric functions y(theta) and x(theta), including sinusoidal and logarithmic components, to compute dy/dx.

This solved example differentiates y and x with respect to theta, then applies dy/dx = (dy/dtheta)/(dx/dtheta) to reveal the derivative of y with respect to x equals 10.

Differentiate x(theta) = 2 cos theta - cos 2 theta and y(theta) = 2 sin theta - sin theta; apply chain rule obtain dy/dx = -1 at theta = pi/2.

This solved example treats x as cos inverse(1/√(1+t^2)) and y as sine inverse(t/√(1+t^2)) of t. It demonstrates that dy/dx = 1.

Differentiate one function with respect to another using du/dv = (du/dx)/(dv/dx), with examples from e^x and √x to sine inverse x and tangent inverse x.

Learn how to compute higher order derivatives, using notations y1, y2, and y3, with examples for x^7 and log x. Solve problems and derive y'' = cos x/(1−sin x)^2.

Analyze solved differential calculus examples by deriving first and second derivatives for y = x log x/(A+Bx) and y = e^(4x) sin 3x, applying product and chain rules.

From scratch, solve key differential calculus problems by deriving dy/dx and the second derivative from parametric equations, and prove that for y = x the nth derivative equals n factorial.

Differentiate y = sin^2 x to obtain y1 and y2, apply chain and implicit differentiation, and manipulate the resulting identities to verify the prescribed relation.

Explore how to compute velocity and acceleration from a straight-line distance function using derivatives with respect to time, and apply these concepts to solve a car's speed problem.

In one-dimensional motion, this lecture derives velocity from displacement, shows sqrt(x) equals twice velocity, and proves acceleration is negative and proportional to the cube root of velocity.

Demonstrates that for a particle moving in a straight line with displacement s proportional to t^n, velocity v and acceleration f satisfy v^2 = n f s/(n-1).

By deriving velocity and acceleration from displacement, this lecture identifies the interval where velocity is negative and acceleration positive, yielding 2 < t < 5.

Solve motion with s = 29.4 t − 4.9 t^2 for a stone thrown upward. Maximum height occurs when velocity is zero at t = 3 s, yielding 44.1 units.

determine velocity of a stone thrown vertically upward to reach maximum height using s = -4.9 t^2 and zero velocity at t = 3 s, yielding u = 29.4 m/s.

Compute the rate of change of circle area (A = pi r^2, dA/dr = 2 pi r) and cylinder's surface-area rate dS/dr = 4 pi r + 2 pi h.

Use A = 4πr^2, so dA/dt = 8πr dr/dt. With r = 4 cm and dr/dt = 0.02 cm/s, dA/dt ≈ 2.01 cm^2 per second.

Apply differential calculus to a sphere by linking dV/dt and dA/dt: find dr/dt from dV/dt=20 and r=8, then compute dA/dt, yielding 5 cm^2/s.

Assumes the cube's volume grows at a constant rate and derives dx/dt = K/(3x^2). Demonstrates dS/dt = 4K/x, so the surface-area change varies inversely with edge length.

Solve for points on the curve 6y = x^3 + 2 where the y-coordinate changes eight times as fast as the x-coordinate; the points are (4, 11) and (-4, -31/3).

Analyze a conical vessel (height 10 m, radius 5 m) filled at 1.5 m^3/min to find dh/dt when h = 4 m, yielding 3/(8 pi) through similar triangles.

Apply Rolle's theorem to a real-valued function on the closed interval [a, b], ensuring a point c in (a, b) with f'(c)=0 when continuity and differentiability hold and f(a)=f(b).

Apply Rolle's theorem to a polynomial on [1,3] to locate c where f'(c)=0, giving c=2±1/√3, and verify it for f(x)=sin(2x) on [0, π/2] with c=π/4.

Apply Rolle's theorem to f(x)=sin(2x) on the interval [0, pi/2], verify continuity and differentiability, show f(0)=f(pi/2)=0, and locate c in (0, pi/2) with f'(c)=0, yielding c=pi/4.

Apply Rolle's theorem to f(x) = e^x cos x on [-pi/2, pi/2], verify continuity and differentiability, show f(-pi/2)=f(pi/2)=0, and find c=pi/4 with f'(c)=0.

Show that the absolute value function |x| is not differentiable at x = 0 on the interval [-1, 1], since the left and right derivatives are -1 and 1.

Lagrange's mean value theorem requires f to be continuous on [a,b] and differentiable on (a,b). There exists c in (a,b) with f'(c) = (f(b)−f(a))/(b−a), so tangent slope matches chord.

Apply Lagrange's mean value theorem to locate the point on y=(x-3)^2 where the tangent parallels the chord between (3,0) and (4,1), yielding (7/2, 1/4).

Verify Lagrange's mean value theorem for two polynomials on [0,1] and [0,4], finding c = 1/2 in (0,1) and c ≈ 2.13 in (0,4).

Apply Lagrange's mean value theorem to f(x)=x^2+1/x on [1,3], showing continuity and differentiability and locating C in (1,3) with f'(C) equal to the secant slope.

Apply Lagrange's mean value theorem to f(x)=1/(4x-1) on [1,4], verify continuity and differentiability, and find c about 1.92 with f'(c) equal to the slope between f(4) and f(1).

Solve for c in the mean value theorem for f(x)=2x^2−10x+29 on [2,7] by computing f(7) and f(2) and equating f'(c) to the slope, yielding c=4.5.

Explore tangents and normals to a curve y = f(x) at a point, using dy/dx as the tangent slope and the perpendicular line as the normal.

Compute derivatives to find tangent and normal lines to polynomial curves, and express their equations at given points.

Differentiate the ellipse x^2/a^2 + y^2/b^2 = 1 to find the slope dy/dx = - (b^2 x)/(a^2 y) and derive the tangent at (x1,y1): x x1 / a^2 + y y1 / b^2 = 1.

For the parametric curve x = a sin^3 t, y = B cos^3 t, compute dy/dx from dy/dt and dx/dt to obtain the tangent and normal at point D.

Differentiate the curve x/a^n + y/b^n = 2 to find the slope of the tangent at (a,b); the tangent equation simplifies to x/a + y/b = 2, independent of n.

Demonstrate that the contact point of tangents from the origin to y = sin x lies on the curve, using tangent slope cos x1 and y1/x1 = cos x1.

Compute the tangent to the curve y = B e^{-x/a} at the y-axis intercept (x = 0) using the derivative, yielding the tangent equation y/B + x/a = 1.

Determine tangent to curve x^m y^n = a^{m+n} via differentiation, locate contact point (x1, y1), and show x cos alpha + y sin alpha = P is tangent.