Question

question 7. 1 point what is the first derivative of ( f(x)=9^{(x^{4})})? select the correct answer below: ( f(x)=(x^{4})9^{(x^{4}-1)}) ( f(x)=(ln 9)9^{(x^{4})}) ( f(x)=(4x^{3})9^{(x^{4})}) ( f(x)=9^{(x^{4})}x^{3}(9ln 4)) ( f(x)=9^{(x^{4})}x^{3}(4ln 9))

Explanation:

Step1: Apply chain - rule

Let $u = x^{4}$, then $y = 9^{u}$. The derivative of $a^{u}$ with respect to $u$ is $a^{u}\ln a$, and the derivative of $u=x^{4}$ with respect to $x$ is $4x^{3}$.

Step2: Calculate derivative

$f'(x)=\frac{d}{dx}(9^{x^{4}})=9^{x^{4}}\ln 9\cdot4x^{3}=(4x^{3})(\ln 9)9^{x^{4}}$

Answer:

$(4x^{3})(\ln 9)9^{x^{4}}$