Applied Math- Limit, Derivative & Applications of Derivative

Learn how derivatives explain daily change and how they keep temperatures stable in devices, air conditioners, and engines, while shaping aerodynamics, manufacturing costs, and reaction rates.

Define the derivative via the first principle, as the limit of (f(x+h) − f(x))/h as h approaches zero, representing the slope of the tangent and rate of change.

Derive standard function derivatives from first principles, deriving sin x as cos x and 2x as 2, and show how derivative formulas originate from the limit definition.

Learn the derivatives of standard functions, including constants, powers, roots, exponentials, and logarithms, with key formulas for x^n, e^x, and ln x; plus derivatives of trigonometric and inverse functions.

Explore derivatives properties: addition, subtraction, constant multiple, product and quotient rules, and the chain rule for composite functions, with examples like sin x and cos x.

Master the chain rule by differentiating nested functions, from roots and sines to logarithms, with step-by-step examples of outer versus inner derivatives.

Learn to differentiate an implicit function with respect to x, collecting dy/dx terms on one side, and apply the method to x^2 + y^2 + x y = 3x.

Learn to differentiate parametric functions by treating x and y as functions of a parameter t, then compute dy/dx as (dy/dt)/(dx/dt) for the slope.

Learn how to write and compute higher order derivatives, from f'(x) to f'''(x), and from dy/dx to d^n y/dx^n, including parametric cases with examples like log(log x).

explore logarithmic differentiation for functions of the form f(x)^{g(x)}. derive the general formula y' = f(x)^{g(x)}[ g'(x) ln f(x) + g(x) f'(x)/f(x) ] and apply to x^x.

Explore the nth derivative of standard functions, including sine and cosine with shifts, powers (x+b)^m, and reciprocals (x+b)^{-1}, with case-based formulas.

Learn to differentiate determinants by differentiating one column (or row) at a time while keeping the rest constant. The video demonstrates applying the rule to determinants with functions inside.

Study Rolle's theorem, requiring continuity on a closed interval, differentiability on the open interval, and equal endpoint values; then find a point c in (a,b) with f'(c)=0.

Understand Lagrange's mean value theorem: a function continuous on [a, b] and differentiable on (a, b) has a c with f'(c) equal to the secant slope between a and b.

Determine the angle of intersection of two curves by using the slopes of their tangents at the intersection and the angle between tangents formula; note orthogonal and touching cases.

Explore derivatives as rate of change, using the limit definition to relate displacement, velocity, and acceleration, and apply rate of change concepts to geometry and volume.

Explore monotonic functions, defined as functions that are totally increasing or decreasing on the interval a to b, including strictly increasing, strictly decreasing, non-decreasing, and non-increasing cases.

Apply derivatives to find tangent and normal lines of a function, using the derivative as slope and the slope point form for tangent and normal equations.

Learn how to determine the lengths of tangent, sub-tangent, normal, and sub-normal to a function at a point, using the derivative and coordinates and applying the relevant formulas.

Solve a basic limit as x approaches zero involving sin x and cos x, using the conjugate method and trigonometric identities to show the limit equals minus one.

Explore how to evaluate a limit at a differentiable point using algebraic manipulation, arrive at the derivative, and compare with L'Hôpital's rule in a state board style question.

Apply L'Hôpital's rule to evaluate the limit by differentiating the numerator and denominator, then substitute x = 8 to obtain the value 5.

Apply a standard limit as x approaches zero by rewriting the expression to match (2x-1)/x, introducing alpha and beta, and multiplying/dividing to complete the pattern, yielding alpha minus beta.

This lecture demonstrates solving a limit problem by substitution to shift the limit to zero, applying limits (2x-1)/x -> 1 and sin x / x -> 1 to obtain -3.

Apply the half-angle formula to evaluate the limit as x approaches 0 of (1 - cos(n x)) / (1 - cos(a n x)), yielding the value 1/a^2.

This lecture solves a limit by using conjugates to remove roots, applying the half-angle identity for 1−cos x, and dividing by x^2 to reach a result expressed with logs.

Apply the standard limit formula to transform the expression, adding and subtracting two to match the pattern, and compute the limit to obtain ((5/3)e + 2)^{2/3}.

Learn to evaluate a zero-limit problem by rewriting in the standard (1+x)^8 form, dividing by x, and applying the limit to obtain a final value of four.

Apply the cos difference identity to rewrite the limit as x approaches zero, separate terms, and use the standard limit sin t / t and L'Hôpital's rule to obtain four.

This lecture demonstrates solving a limit as h approaches zero using the sin a minus b formula and standard trigonometric limits, yielding four by three.

This lecture teaches techniques to solve limits of rational and root functions using factoring, cancellation, and conjugates, with substitution for zero in trigonometric limits and synthetic division when needed.

Explore derivatives at level one and apply the quotient rule to a trigonometric function, differentiating the numerator and denominator and simplifying to the final expression.

Apply logarithmic differentiation to y = x^{sin x} by taking log y = sin x log x and differentiating. Substituting x = pi/2 yields dy/dx = 1 and y(pi/2) = pi/2.

Apply the inverse derivative formula to find the derivative of arctan of the square root, using the chain rule, yielding 1/(2 sqrt(x)(1+x)).

Learn to compute the second-order derivative with respect to x for a parametric function by first finding dy/dx and then differentiating with respect to x.

Apply log differentiation to find the derivative of y = (1+4x)^5(3+x-x^2)^2, giving dy/dx = (1+4x)^5(3+x-x^2)^2 [20/(1+4x) + 2(1-2x)/(3+x-x^2)].

apply the cos triple-angle identity to simplify the expression, convert to sine inverse to cancel, and differentiate to get -6.

Reframe the expression with the cos 2θ identity, substitute θ = 3x, and differentiate via the chain rule, including the derivative of 3^{2x} using logarithms.

solve a derivative problem by converting logarithmic expressions to exponential form, cross-multiplying to collect terms, and differentiating to obtain dy/dx, yielding a final simplified ratio.

Solves the infinite nested radical y = sqrt(x + y) by squaring and implicit differentiation. Derives dy/dx, yielding dy/dx = sin x / (1 - 2y).

learn implicit differentiation to solve a derivative problem on the curve defined by (x/a)^{2/3} + (y/a)^{2/3} = 1, and obtain dy/dx = -((x/a)^{1/3})/((y/a)^{1/3}).

Differentiate a logarithmic expression by applying log properties to simplify products and quotients. Differentiate term by term and simplify the resulting expression.

Explore applied derivative techniques through solved numericals: tangent and normal equations, related rates with shadows, rate of volume change from surface area, linear approximation of functions, and Rolle's theorem applications.