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hw14 the chain rule (target c4; §3.6) due: thu oct 9, 2025 11:59pm
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hw14 the chain rule (target c4; §3.6) score: 8/11 answered: 8/11
question 9
find the derivative of: -7 sin²(5x⁹).
hint: sin²(x)=sin(x)²...so use the chain rule (twice!).
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Explanation:

Step1: Let $u = 5x^{9}$

Let $y=-7\sin^{2}(u)$. First - use the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$.

Step2: Differentiate $y = - 7\sin^{2}(u)$ with respect to $u$

Let $v=\sin(u)$, so $y = - 7v^{2}$. First, find $\frac{dy}{dv}$ and $\frac{dv}{du}$.
$\frac{dy}{dv}=-14v$ (using the power rule $\frac{d(ax^{n})}{dx}=nax^{n - 1}$ with $a=-7$ and $n = 2$).
$\frac{dv}{du}=\cos(u)$ (since $\frac{d\sin(u)}{du}=\cos(u)$).
Then, by the chain - rule $\frac{dy}{du}=\frac{dy}{dv}\cdot\frac{dv}{du}=-14v\cos(u)=-14\sin(u)\cos(u)$.

Step3: Differentiate $u = 5x^{9}$ with respect to $x$

Using the power rule $\frac{d(ax^{n})}{dx}=nax^{n - 1}$, where $a = 5$ and $n = 9$, we get $\frac{du}{dx}=45x^{8}$.

Step4: Apply the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$

Substitute $\frac{dy}{du}=-14\sin(u)\cos(u)$ and $\frac{du}{dx}=45x^{8}$ into the chain - rule formula. Since $u = 5x^{9}$, we have:
$\frac{dy}{dx}=-14\sin(5x^{9})\cos(5x^{9})\cdot45x^{8}=-630x^{8}\sin(5x^{9})\cos(5x^{9})$.
Using the double - angle formula $\sin(2\theta)=2\sin\theta\cos\theta$, we can rewrite it as $\frac{dy}{dx}=-315x^{8}\sin(10x^{9})$.

Answer:

$-315x^{8}\sin(10x^{9})$