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 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).

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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

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.

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).

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.