Question

Which of the following represents the derivative of

⁡ ln(cos^2(x)) with respect to x?

A) −2tan(x)

B) tan(x)

C) −tan(x)

D) 2tan(x)

Answer 1

The derivative of ln(cos^2(x)) with respect to x is found using the chain rule and trigonometric identities, resulting in -2tan(x), which is answer choice A.

Step-by-step explanation:

The derivative of ℓ(cos2(x)) with respect to x can be found using the chain rule and trigonometric identities. First, apply the chain rule: the derivative of ℓ(u) with respect to u is 1/u, multiplied by the derivative of u with respect to x. Here, u = cos2(x).

The derivative of cos2(x) with respect to x is 2cos(x)(-sin(x)) by using the power rule and the derivative of cosine, which is -sin(x). Combining these results, we have 1/cos2(x) × -2cos(x)sin(x).

Using the trigonometric identity sin(2x) = 2sin(x)cos(x), we can simplify to -sin(2x)/cos2(x). Recognizing that sin(2x)/cos2(x) is the same as sin(2x)/cos(x)× 1/cos(x), which simplifies to 2sin(x), we then use the identity for tangent, tan(x) = sin(x)/cos(x), yielding -2tan(x).

Therefore, the derivative of ℓ(cos2(x)) with respect to x is -2tan(x), which corresponds to answer choice A.