Question

find the derivative of: 7 sin²(-9x⁸). hint: sin²(x) = sin(x)²...so use the chain rule (twice!).

Explanation:

Step1: Let \(u = - 9x^{2}\)

Let \(y = 7\sin^{2}(u)\)

Step2: Apply the chain rule

The derivative of \(y\) with respect to \(x\) is \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\)
First, find \(\frac{dy}{du}\):
Using the power - rule and the derivative of sine function. If \(y = 7\sin^{2}(u)=7(\sin(u))^{2}\), then by the power - rule \((X^{n})^\prime=nX^{n - 1}\) and \((\sin(u))^\prime=\cos(u)\)
\(\frac{dy}{du}=7\times2\sin(u)\cos(u)=14\sin(u)\cos(u)\)
Second, find \(\frac{du}{dx}\):
If \(u=-9x^{2}\), then \(\frac{du}{dx}=-18x\)

Step3: Substitute back

\(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=14\sin(-9x^{2})\cos(-9x^{2})\cdot(-18x)\)
Since \(\sin(A)\cos(A)=\frac{1}{2}\sin(2A)\), we can also write it as \(-126x\sin(2(-9x^{2}))=-126x\sin(-18x^{2})\)
Another way: Using the double - angle formula \(\sin(2\theta) = 2\sin\theta\cos\theta\), \(\frac{dy}{dx}=-126x\cos(-9x^{2})\sin(-9x^{2})\)

Answer:

\(-126x\cos(-9x^{2})\sin(-9x^{2})\)