Question

For f(x) = cos(1 + 2x), what is f '(x)?

A) f '(x) = −2 sin(1 + 2x)
B) f '(x) = −sin(1 + 2x)
C) f '(x) = sin(1 + 2x)
D) f '(x) = 2 sin(1 + 2x)

Answer 1

To find the derivative of f(x) = cos(1 + 2x), the chain rule is applied to the composition of the cosine function and linear function 1 + 2x, resulting in f'(x) = -2 sin(1 + 2x), which is answer choice A.

Step-by-step explanation:

To find the derivative of f(x) = cos(1 + 2x), we need to use the chain rule of differentiation. The chain rule states that if a function y = f(g(x)), then its derivative is dy/dx = f'(g(x)) · g'(x).

In our case, the outer function is the cosine function, and the inner function is the linear function 1 + 2x.

First, we find the derivative of the outer function. The derivative of cos(u) with respect to u is -sin(u).

Then, we differentiate the inner function 1 + 2x, which is simply 2 since the derivative of a constant is 0, and the derivative of 2x with respect to x is 2.

Applying the chain rule, we multiply the derivatives of the outer and inner functions:

f'(x) = -sin(1 + 2x) · 2

Thus, the derivative of f(x) is:

f'(x) = -2 sin(1 + 2x)

The correct answer to the question is A) f'(x) = −2 sin(1 + 2x).