Question

section 2.6: chain rule (homework)
score: 40/170 answered: 4/17
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question 5
0/10 pts 4 99 details
use the chain rule to find the derivative of 2e^(-3x^9 + 2x^3)
use e^x for e^x.
enter an algebraic expression more...

Explanation:

Step1: Recall the chain - rule formula

The chain - rule states that if $y = f(g(x))$, then $y^\prime=f^\prime(g(x))\cdot g^\prime(x)$. Also, the derivative of $e^u$ with respect to $x$ is $e^u\cdot u^\prime$, where $u$ is a function of $x$, and the derivative of a constant times a function is the constant times the derivative of the function. Let $y = 2e^{-3x^{9}+2x^{3}}$, and let $u=-3x^{9}+2x^{3}$. Then $y = 2e^u$.

Step2: Find the derivative of the outer function

The derivative of $y = 2e^u$ with respect to $u$ is $\frac{dy}{du}=2e^u$.

Step3: Find the derivative of the inner function

The derivative of $u=-3x^{9}+2x^{3}$ with respect to $x$ is $\frac{du}{dx}=-3\times9x^{8}+2\times3x^{2}=-27x^{8} + 6x^{2}$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=2e^u$ and $\frac{du}{dx}=-27x^{8}+6x^{2}$ into the chain - rule formula. Since $u=-3x^{9}+2x^{3}$, we have $\frac{dy}{dx}=2e^{-3x^{9}+2x^{3}}\cdot(-27x^{8}+6x^{2})$.

Answer:

$2(-27x^{8}+6x^{2})e^{-3x^{9}+2x^{3}}$