Question

Let sin(h(x)) be the function defined by sin(h(x)), where h(x) is a differentiable function. Which of the following is equivalent to the derivative of sin(h(x)) with respect to x?

a) h′(x)cos(h(x))
b) sin′(h(x))⋅h′(x)
c) cos(h(x))⋅h′(x)
d) sin(h(x))⋅h′(x)

Answer 1

The derivative of sin(h(x)) with respect to x is cos(h(x)) ⋅ h′(x), which is option c. The chain rule of differentiation is applied here, where the derivative of the outer function (sine) is taken and multiplied by the derivative of the inner function h(x).

Step-by-step explanation:

To find the derivative of sin(h(x)) with respect to x, we need to apply the chain rule of differentiation. This rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In this case, the outer function is the sine function and the inner function is h(x).

The derivative of sin(u) where u is a function of x, is cos(u) times the derivative of u. Therefore, the derivative of sin(h(x)) with respect to x is cos(h(x)) ⋅ h′(x). This matches option c in the choices you've provided.

The incorrect options can be ruled out as follows:

• Option a (h′(x) cos(h(x))) incorrectly suggests that the derivative of h(x) should be on the left and does not show that it's the product with the cosine function.
• Option b (sin′(h(x)) ⋅ h′(x)) is not following the differentiation of the sine function.
• Option d (sin(h(x)) ⋅ h′(x)) mistakenly keeps the sine function instead of using its derivative.