Finding Derivative Values


In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.


f(u) = u + 1/cos²u, u = g(x) = πx, x = 1/4

Verified step by step guidance

1. First, understand that you need to find the derivative of the composite function (f ∘ g)(x), which is f(g(x)).
2. Apply the chain rule for derivatives, which states that (f ∘ g)'(x) = f'(g(x)) * g'(x).
3. Calculate g'(x) for g(x) = πx. Since g(x) is a linear function, g'(x) = π.
4. Next, find f'(u) for f(u) = u + 1/cos²u. Use the derivative rules: the derivative of u is 1, and for 1/cos²u, apply the chain rule and the derivative of trigonometric functions.
5. Evaluate (f ∘ g)'(x) at x = 1/4 by substituting g(1/4) into f'(u) and multiplying by g'(1/4).

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. If you have a function h(x) = f(g(x)), the derivative h'(x) is found by multiplying the derivative of the outer function f at g(x) by the derivative of the inner function g at x. This is essential for finding (f ∘ g)'(x).

Derivative of Trigonometric Functions

Understanding how to differentiate trigonometric functions is crucial. For example, the derivative of cos(u) is -sin(u), and using the chain rule, the derivative of 1/cos²(u) involves applying the power rule and the derivative of cosine. This knowledge is necessary to differentiate f(u) = u + 1/cos²(u).

Substitution

Substitution involves replacing variables with given values to simplify the differentiation process. In this problem, you substitute u = g(x) = πx and x = 1/4 into the functions. This step is crucial for evaluating the derivative at the specific point x = 1/4, ensuring accurate computation of (f ∘ g)'(x).