Mathematics Journey into Calculus for machine learning

Explore the derivative at a point using the limit definition on an open interval. Define f'(c) as the limit of (f(x)-f(c))/(x-c) with finite value, and equal left and right limits.

Compute the derivative of a constant function f(x)=k using the limit definition, showing it equals zero at x=0 and at x=5.

Apply the limit definition to f(x)=x^2+1 and find its derivative at x=2, showing f'(2)=4 for machine learning.

Apply the limit definition to differentiate f(x) = x^2 + 4x + 3. Obtain f'(1) = 6 and f'(3) = 10.

Compute the derivative of f(x)=2x^2+3x-5 at x=-1 and find f(0); then prove that f(0) + 3 f'(-1) = 0, by evaluating limits and derivatives.

Understand the derivative as instantaneous speed at a time, linking distance f(t) to time and using the limit h→0, e.g., f(t)=t^2+3t+4 gives 7 m/s at t=2.

Understand the geometrical interpretation of the derivative at a point as the slope of the tangent to the curve, defined by the limit of (f(c+h)-f(c))/h as h approaches zero.

Learn to derive the derivative from first principles using the limit definition and delta method. Explore key limits and e^x expansion that underpin differentiation from first principles, including log limits.

Derive x^n using the first principle and the limit definition, arriving at the power rule dy/dx = n x^{n-1}, with examples like x^5 and 1/x^3.

Learn basic formulae for finding the differential coefficient, including power, exponential, and logarithmic rules, and apply them to solve problems.

Differentiate several functions with respect to x, including log base x, e-based expressions, and exponentials. Apply log properties, base considerations, and power rules to compute dy/dx.

Master essential differentiation rules, including constant multiple, sum, difference, product, and quotient rules, with examples such as x cubed and log x upon x square.

Differentiating several functions with respect to x, the lecture applies the sum rule, power rule, and index rules, and simplifies expressions involving x powers and root x.

Derive that the series y = 1 + x/1! + x^2/2! + ... equals e^x, and show that dy/dx = e^x = y, illustrating the self-derivative property of the exponential.

The lecture demonstrates that for y = sum_{k=0}^n x^k/k!, differentiating term by term yields dy/dx = y - x^n/n!, hence dy/dx - y + x^n/n! = 0.

Differentiate two functions with respect to x: first, a quadratic product (x^2 - 3x + 2)(x + 2); second, (x^2 + 1/x^2)^3, using the power rule and binomial expansion.

Differentiate f(x) = alpha x^n to get f'(x) = alpha n x^{n-1}, substitute x = 1 to obtain F-1 = alpha n, hence alpha = F-1 / n.

Explains the derivative of x^n, using f'(x)=n x^{n-1}, evaluating at x=1 to get f'(1)=n, and with F-1=10 concludes n=10.

solve a linear function problem where f(x) = m x + c, with f(0) = 1 and f'(0) = 1, to determine m, c, and f(2) = 3.

Differentiate y applying the product rule to e^x times (x plus log x) and treat y as a triple product x^n log x base a times e^x to find dy/dx.

Apply the product rule to differentiate y = x^3 e^x + (x^2+1)(x-1) with respect to x, derive dy/dx, and present the simplified result.

Apply the quotient rule to differentiate y = (2x+3)/(x^2-5) and y = (x+e^x)/(1+log x base e), derive and simplify dy/dx.

Explore how dy/dx and dx/dy are reciprocals, with their product equal to one, and how differentiation of implicit functions is introduced.

Learn the chain rule, the differentiation of a function of a function. Apply dz/dx = (dz/dy)(dy/dx) for z=f(y) and y=g(x), as in y=(x^2+1)^2, where dy/dx=4x(x^2+1).

Apply the chain rule to differentiate (x^2 + x + 1)^4, sqrt(x^2 + x + 1), and (a^2 − x^2)^{−1/2}, with step-by-step guidance on inner and outer derivatives.

Example 2 differentiates functions: y = e^{e^x} with dy/dx = e^{e^x} e^x, and y = log_7 x with dy/dx = 1/(x log 7), illustrating the chain rule and logarithms.

Apply chain rule to y = (x + sqrt(x^2 + a^2))^n; derivative of inside is 1 + x/ sqrt(x^2 + a^2); obtain dy/dx = n y / sqrt(x^2 + a^2).

Explore differentiation in calculus for machine learning: log x base ten and log ten base e, using quotient and power rules, and differentiate a five-power expression via the chain rule.

Learn to differentiate implicit functions, when not possible to write y in terms of x, by differentiating f(x,y)=0 with respect to x using dy/dx.

Apply implicit differentiation to x^2 + 2xy + y^3 = 42 using product and chain rules, and solve for dy/dx as (-2x + y)/(2x + 3y^2).

Differentiate implicitly the equation x^3 + y^3 = 3 a x y with respect to x, apply the product rule, and solve for dy/dx.

Convert the implicit equation to an explicit function by transposing and squaring, yielding y = -x/(1+x) for x ≠ y. Differentiate to get dy/dx = -1/(1+x)^2.

Apply logarithmic differentiation to f(x)^{g(x)} such as x^x or x^y by taking logs and differentiating to obtain dy/dx in terms of f, g, and their derivatives.

Differentiate exponential functions like x^x and x^√x using logarithmic differentiation, applying product and chain rules to derive dy/dx.

Differentiate y = 5^x / x^5 using logarithmic differentiation, yielding dy/dx = (5^x / x^5) (log 5 − 5/x). Cross-check the result using the quotient rule.

Learn to differentiate a product of powers using logarithmic differentiation; take logs and obtain dy/dx = y[2/(x+1) + 3/(x+2) + 4/(x+3)].

Use logarithmic differentiation to solve for y in the equation x^y = e^{x - y} and derive dy/dx. The result is dy/dx = log x / (log x + 1)^2.

Derive dy/dx for x^y = y^x by taking logs and applying the product rule, yielding an expression in x and y.

Apply logarithms to both sides of x^m y^n = (x+y)^{m+n}, differentiate with respect to x, and solve for dy/dx. Derive dy/dx = y/x.

This lecture uses log differentiation to differentiate a rational y and shows that y/x equals a/(a−x) + b/(b−x) + c/(c−x).

Prove that the derivative of an even function is odd, and the derivative of an odd function is even, with examples using x^2 and x^3.

Explore differentiation of infinite series and differentiate y = sqrt(x + y) with respect to x when the series extends to infinity.

Investigate the infinite nested radical y = sqrt(x + sqrt(x + ...)) and prove via squaring and differentiation that dy/dx = 1/(2y - 1).

Differentiate the infinite power tower y = x^y by taking logs, apply the product rule, and obtain dy/dx = y^2 / (x(1 - y log x)).

Differentiate the infinite continued fraction defining y, using cross multiplication and implicit differentiation, to obtain dy/dx = y/(2y minus x).

Derive dy/dx for the infinite sum y = e^x + y by taking logs: ln y = x + y, differentiate to obtain dy/dx = y/(1−y), and verify the relationship.

Learn to differentiate parametric functions by differentiating x(t) and y(t) and computing dy/dx as (dy/dt)/(dx/dt), or eliminate t to relate x and y.

Compute dy/dx for y = a t^2 and x = 2 a t by dy/dt divided by dx/dt, then eliminate t to get dy/dx = x/(2a) and y = x^2/(4a).

Derive dx/dt and dy/dt for x=(1+t^2)/(1-t^2) and y=2t/(1-t^2) using quotient rule, then compute dy/dx as (dy/dt)/(dx/dt), yielding (1+t^2)/(2t).

Differentiate x with respect to t and y with respect to t, then form dy/dx as (dy/dt)/(dx/dt). Using x=(e^t+e^-t)/2 and y=(e^t-e^-t)/2, dy/dx = x/y.

Discover how to differentiate a function with respect to another function, using the formula du/dv = (du/dx)/(dv/dx), with examples like differentiating x^2 with respect to x^3.

Differentiate x^2 with respect to x^3 by setting u = x^2 and v = x^3, then use du/dv = (du/dx)/(dv/dx) to get 2/(3x).

Explore higher order derivatives, standard notations for dy/dx, d2y/dx2, and beyond, and relate them to velocity and acceleration, using the sign of the second derivative to determine maxima or minima.

Demonstrate that y = a e^{m x} + b e^{n x} satisfies the differential equation d^2y/dx^2 − (m+n) dy/dx + mn y = 0 by differentiating and substituting.

Derive y = x log x /(a+ b x) using quotient and chain rules, apply log properties, and prove x^3 y'' = (x y' - y)^2.

Apply log differentiation to y = x^x, derive dy/dx = x^x(1+log x), and show d2y/dx2 = x^x(1/x + 1 + log x)^2.

Differentiate y = log x + sqrt(x^2 + a^2) to find dy/dx and, after squaring and differentiating, derive the equation (x^2 + a^2) y'' + x y' = 0.

Explore rate of change in calculus—from average rate to differential coefficient dy/dx—and show its use in physics, motion, commerce, and machine learning, including velocity, acceleration, and marginal cost and revenue.

Compute rate of change of circle area with respect to radius; dA/dr = 2 pi r, giving 6 pi and 8 pi for r = 3 cm and 4 cm.

Use chain rule on a cube with v = x^3 and S = 6x^2; with dv/dt = 8, find ds/dt when x = 12 cm, yielding ds/dt = 8/3 cm^2/s.

The cube's edge grows at 10 cm/s, with V = x^3, so dv/dt = 3x^2 dx/dt, giving 750 cm^3/s at x = 5 cm.

Compute dV/dx for a sphere whose diameter is d = (3/2)(2x+3). With r = d/2 and V = 4/3 pi r^3, dV/dx = (27/8) pi (2x+3)^2.

Differentiate A = pi r^2 to relate da/dt and dr/dt. Substitute r = 7.5 cm and dr/dt = 3.5 cm/s to yield da/dt = 52.5 pi cm^2/s.

From x=2t^3-9t^2+5t+8, differentiate to obtain velocity 6t^2-18t+5 and acceleration 12t-18, find t=3/2 s where acceleration is zero, and velocity -8.5 cm/s.

Apply the rate of change concept by differentiating the cost function to obtain marginal cost, then evaluate at x = 17 units to find MC about 20.97 rupees.

Explore tangents and normals on curves, derive slope via dy/dx, and relate these to gradients in machine learning, while reviewing line equations and coordinate geometry basics.

Differentiate implicitly to obtain dy/dx = -2x/(3+2y) for the curve x^2+3y+y^2=5; at (1,1) the tangent slope is -2/5 and the normal slope is 5/2.

Compute dy/dx = 6x^2 for y = 2x^3 − 3; at x = 2 and x = -2 the slope is 24, so the tangents are parallel.

Differentiate y = x^2 - 5x + 6: dy/dx = 2x - 5; at x=2, m1 = -1 and at x=3, m2 = 1; m1m2 = -1, tangents are perpendicular.

Differentiate the curve 2 y^2 = a x^2 + b; at (1, -1) with slope -1, solve to find a and b (a = 2, b = 0).

Differentiate y = x^3 − 2x^2 − x and set dy/dx = 3 to find tangents parallel to y = 3x − 2; x = 2 or −2/3 yield (2, −2) and (−2/3, −14/27).

Identify the point on the curve y = 2x^2 - 6x - 4 where the tangent is parallel to the x-axis by solving dy/dx = 4x - 6 = 0, yielding (3/2, -17/2).

Differentiate the curve w.r.t. y and set dx/dy to zero for tangents parallel to the y axis; y = 2 yields x = -1 or 3, (-1,2) and (3,2).

Explore points on the curve y = x^3 where the tangent slope equals the point's y-coordinate, solving dy/dx = 3x^2 = x^3 to find (0,0) and (3,27).

Analyze the hyperbola x^2/9 - y^2/16 = 1 to locate tangents parallel to the axes, finding vertical tangents at (±3,0) and showing no horizontal tangent exists.

Identify a point on the curve y = (x−3)^2 where the tangent is parallel to the line through (4,1) and (3,0); the caption gives the point (3, 1/4).

Introduce integration as the reverse of differentiation and present basic formulae for x^n, e^x, and a^x, including the constant of integration and indefinite versus definite integrals.

Apply the power rule for integrals to evaluate ∫ x^4 dx, ∫ x^(1/2) dx, ∫ x^-3 dx, and transform log base a of x^3 into ∫ x^3 dx.

Compute the integration of x^n terms using x^(n+1)/(n+1), yielding x^4/4, 5/3 x^3, -4x, 7 log modulus x, 4 sqrt x, and c.

Divide each term by x^2 and integrate term by term. Obtain x^2/2 + 5x + 4 log|x| - 1/x + c; for the second integral, (2/3)x^(3/2) - (2/5)x^(5/2) + c.

expand (√x + 1/√x)^2 to x + 1/x + 2 and integrate to x^2/2 + log|x| + 2x + c, noting ∫ a^x dx = a^x / log base e a.

Explain and apply the exponential integral formula for a^x to (2/5)^x+(3/5)^x, then perform polynomial division of (x^4+x^2+1)/(x^2−x+1) to integrate x^2+x+1.

Explore fundamental calculus concepts used in business, including the cost function, revenue, profit, break-even analysis, and marginal cost and marginal revenue, with applications to profit optimization.

Compute break-even points by setting profit p(x)=r(x)−c(x) to zero, with r(x)=10x−0.2x² and c(x)=x+40; solving yields x=5 or x=40.

Set profit to zero to find the break-even point. With fixed cost 16,100, variable cost 20 per unit, and price per unit of 27 rupees, break-even occurs at 2,300 units.

Compute the average cost C(x)/x for C(x) = 1500 + 30x + x^2, and derive the marginal cost mc = 30 + 2x, with mc(20) = 70.

Derive the cost function from the average cost, then differentiate to obtain the marginal cost function; evaluate the marginal cost at 100 units to get rupees three.

Differentiate r=t^3 i+2 t^2 j−3 t k to get dr/dt=3 t^2 i+4 t j−3 k and d^2 r/dt^2=6 t i+4 j.

this lecture demonstrates differentiation of r(t)=a cos ωt + b sin ωt, computes dr/dt, uses cos^2+sin^2 to relate a and b in the cross product, and proves d^2r/dt^2 = -ω^2 r.

For a vector function a(t) with constant magnitude, differentiate a(t)·a(t)=a^2 to obtain a(t)·(da/dt)=0, proving the derivative of the squared magnitude is zero.

Compute the derivative of the dot product r · da/dt using the product rule for vectors: d/dt(r · da/dt) equals r · da/dt plus (da/dt) · (da/dt).

Apply the product rule to derivative of r × s with s = dr/dt; show (dr/dt) × (dr/dt) = 0; obtain d/dt(r × dr/dt) = r × d^2 r/dt^2.

Integrate dr/dt equals a t plus b, where a and b are constant vectors, to obtain r equals 1/2 a t^2 plus b t plus c.

Compute the definite integral of the vector r = (t − t^2)i + (2t^3)j − 3tk from t = 2 to 3, producing (-23/6)i + (65/2)j − 3k.

Compute the integral from 2 to 3 of r(t) · dr/dt dt by applying that the square of a vector equals the square of its modulus, yielding (|r(3)|^2−|r(2)|^2)/2 = 10.

Compute the definite integral of the vector e^t i + t^2 j - e^{-2t} k from 1 to 2, yielding the vector with components e^2 - e, 7/3, and (1/2)(e^{-4} - e^{-2}).

Explore basic trigonometry concepts and formulae, including right angle triangle ratios, identities, transformation formulas, and allied-angle signs, with applications to differentiation and integration in machine learning.

Explore illustrated trigonometric identities, proving equalities with sec^2 theta = 1 + tan^2 theta, factorization, and double-angle formulas to derive cos theta over one minus sine theta and sin 3a.

Master the differentiation formulae for trigonometric and inverse trigonometric functions, including sin, cos, tan, cot, sec, csc, and the derivatives of their inverse functions.

Master differentiation techniques for exponential, trigonometric, and composite functions using chain rule, product rule, and quotient rule with examples like e^(sin x) and tan(3x+1).

Explore the formulae for integrating trigonometric and inverse trigonometric functions, including sine, cosine, tangent, secant, cosecant, and log terms, with problem-solving applications.

Solve formula-based integrals involving sine, cosine, tangent, secant, and their squares using trigonometric identities to find antiderivatives and constants of integration.