Questions: What is the derivative of (7^sin (x)) ? (7^sin (x) cdot cos (x)) (sin (x) cdot 7^sin (x)-1) (7^sin (x) cdot ln (7)) (7^sin (x) cdot ln (7) cdot cos (x)) (7^sin (x)) Select each function below that requires the use of the chain rule to find the derivative. (f(x)=(cos (x))^3) (f(x)=x^3 cos (x)) (f(x)=3 cos (x)) (f(x)=x^3+cos (x)) (f(x)=cos left(x^3right))

What is the derivative of (7^sin (x)) ?
(7^sin (x) cdot cos (x))
(sin (x) cdot 7^sin (x)-1)
(7^sin (x) cdot ln (7))
(7^sin (x) cdot ln (7) cdot cos (x))
(7^sin (x))

Select each function below that requires the use of the chain rule to find the derivative.
(f(x)=(cos (x))^3)
(f(x)=x^3 cos (x))
(f(x)=3 cos (x))
(f(x)=x^3+cos (x))
(f(x)=cos left(x^3right))

Transcript text: What is the derivative of $7^{\sin (x)}$ ? $7^{\sin (x)} \cdot \cos (x)$ $\sin (x) \cdot 7^{\sin (x)-1}$ $7^{\sin (x)} \cdot \ln (7)$ $7^{\sin (x)} \cdot \ln (7) \cdot \cos (x)$ $7^{\sin (x)}$ Select each function below that requires the use of the chain rule to find the derivative. $f(x)=(\cos (x))^{3}$ $f(x)=x^{3} \cos (x)$ $f(x)=3 \cos (x)$ $f(x)=x^{3}+\cos (x)$ $f(x)=\cos \left(x^{3}\right)$

Solution

Solution Steps

Solution Approach

To find the derivative of $7^{\sin(x)}$, use the chain rule and the fact that the derivative of $a^{u(x)}$ is $a^{u(x)} \cdot \ln(a) \cdot u'(x)$, where $a$ is a constant and $u(x)$ is a function of $x$.
For the second question, identify functions that require the chain rule. The chain rule is needed when differentiating a composite function, i.e., a function of a function.

Step 1: Derivative of $7^{\sin(x)}$

To find the derivative of the function $f(x) = 7^{\sin(x)}$, we apply the chain rule. The derivative is given by:

\[ f'(x) = 7^{\sin(x)} \cdot \ln(7) \cdot \cos(x) \]

Step 2: Functions Requiring the Chain Rule

Next, we identify which of the given functions require the use of the chain rule for differentiation. The functions are: