9. (2 points) find the derivative of each function. pick 2.\na. $f(x)=x^{2}\tan x$\nb. $g(x)=sin^{7}(3x)$\nc…

Explanation:

Step1: Use product - rule for (f(x))

The product - rule states that if (y = u\cdot v), then (y^\prime=u^\prime v + uv^\prime). For (f(x)=x^{2}\tan x), let (u = x^{2}) and (v=\tan x). We know that (u^\prime = 2x) and (v^\prime=\sec^{2}x). Then (f^\prime(x)=(x^{2})^\prime\tan x+x^{2}(\tan x)^\prime=2x\tan x + x^{2}\sec^{2}x).

Step2: Use chain - rule for (g(x))

The chain - rule states that if (y = f(u)) and (u = g(x)), then (y^\prime=f^\prime(u)\cdot g^\prime(x)). For (g(x)=\sin^{7}(3x)), let (u = \sin(3x)), so (g(x)=u^{7}). First, (\frac{dg}{du}=7u^{6}), and for (u=\sin(3x)), (\frac{du}{dx}=3\cos(3x)). Then (g^\prime(x)=7\sin^{6}(3x)\cdot3\cos(3x)=21\sin^{6}(3x)\cos(3x)).

Answer:

For (f(x)=x^{2}\tan x), (f^\prime(x)=2x\tan x + x^{2}\sec^{2}x) For (g(x)=\sin^{7}(3x)), (g^\prime(x)=21\sin^{6}(3x)\cos(3x))