Prove, using the definition of derivative, that if f(x)=cosxt then f^'(x)=-sinx .

Name: Problem 20, Chapter 3: Calculus Early Transcendentals
Uploaded: 2021-08-10T11:59:02-08:00
Duration: 3 min 27 s
Channel: Ian Grigsby
Description: Prove, using the definition of derivative, that if f(x)=cosxt then f^'(x)=-sinx .

step 1 All right, so we are deriving cosine via the limit definition of a derivative. So what we need t

Prove, using the definition of derivative, that if...

Key Concepts

Standard_Trigonometric_Limits

A key component in the analysis of trigonometric functions is the limit lim??0 (sin h)/h = 1. This fundamental limit is widely employed to evaluate expressions after substitution from trigonometric identities and is critical in establishing the derivative of trigonometric functions.

Differentiability_of_Trigonometric_Functions

Combining the definition of the derivative with trigonometric identities and standard limits provides a rigorous framework to differentiate functions like cosine. By expressing the difference quotient in terms of sine and applying the limit properties, one can deduce that the derivative of cos x is -sin x, thus demonstrating the differentiable nature of trigonometric functions.

Definition_of_Derivative

The derivative of a function at a point is defined as the limit of the difference quotient, specifically, f'(x) = lim??0 [f(x+h) - f(x)]/h. This foundational concept in calculus formalizes the idea of an instantaneous rate of change, serving as the basis for differential calculus and enabling the computation of slopes of tangent lines to curves.

Trigonometric_Identities

Trigonometric identities, such as the cosine difference formula, are invaluable for simplifying expressions involving trigonometric functions. Using identities like cos(a) - cos(b) = -2 sin((a+b)/2) sin((a-b)/2) allows one to rewrite and manage the algebraic complexity that arises when taking limits of expressions involving cosine functions.

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