Question

hw14 the chain rule (target c4; §3.6) score: 8/11 answered: 9/11 question 9 score on last try: 0 of 1 pts. see details for more. next question get a similar question you can retry this ques find the derivative of: -7 sin²(5x⁹). hint: sin²(x)=sin(x)²...so use the chain rule (twice!). -315x⁸ sin(10x⁹) i question help: video message instructor

Explanation:

Step1: Let \(u = 5x^{9}\) and \(y=-7\sin^{2}(u)\)

First - level chain - rule setup.

Step2: Differentiate \(y\) with respect to \(u\)

Using the power - rule and the derivative of \(\sin\) function. If \(y = - 7[\sin(u)]^{2}\), let \(v=\sin(u)\), then \(y=-7v^{2}\). \(\frac{dy}{dv}=-14v\) and \(\frac{dv}{du}=\cos(u)\). So \(\frac{dy}{du}=\frac{dy}{dv}\cdot\frac{dv}{du}=-14\sin(u)\cos(u)= - 7\sin(2u)\) (using the double - angle formula \(\sin(2\alpha)=2\sin\alpha\cos\alpha\)).

Step3: Differentiate \(u\) with respect to \(x\)

If \(u = 5x^{9}\), then \(\frac{du}{dx}=45x^{8}\) by the power rule \(\frac{d}{dx}(ax^{n})=nax^{n - 1}\).

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

Substitute \(u = 5x^{9}\) into \(\frac{dy}{du}\) and multiply by \(\frac{du}{dx}\). \(\frac{dy}{dx}=-7\sin(2\cdot5x^{9})\cdot45x^{8}=-315x^{8}\sin(10x^{9})\)

Answer:

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