Question

hw11 differentiation rules ii (target c1, c2, c5; §3.3)
score: 2/8 answered: 2/8
question 3
use the product rule to find the derivative of (-3x^6 + 2x^10)(10e^x - 2)
use e^x for e^x. you do not need to expand out your answer.
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Explanation:

Step1: Recall product - rule

The product rule states that if $y = u\cdot v$, then $y^\prime=u^\prime v + uv^\prime$. Let $u=-3x^{6}+2x^{10}$ and $v = 10e^{x}-2$.

Step2: Find $u^\prime$

Differentiate $u=-3x^{6}+2x^{10}$ with respect to $x$. Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we get $u^\prime=-18x^{5}+20x^{9}$.

Step3: Find $v^\prime$

Differentiate $v = 10e^{x}-2$ with respect to $x$. Since $\frac{d}{dx}(e^{x})=e^{x}$ and $\frac{d}{dx}(c)=0$ (where $c$ is a constant), we have $v^\prime = 10e^{x}$.

Step4: Apply product - rule

$y^\prime=u^\prime v+uv^\prime=(-18x^{5}+20x^{9})(10e^{x}-2)+(-3x^{6}+2x^{10})(10e^{x})$

Answer:

$(-18x^{5}+20x^{9})(10e^{x}-2)+(-3x^{6}+2x^{10})(10e^{x})$