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

Answer:

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

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}))