Question

In Exercises \(13-24,\) find \(d y / d x .\) If you are unsure of your answer, use NDER to support your computation. $$y=\left(1+\cos ^{2} 7 x\right)^{3}$$

Short answer

Hence, the derivative of the given function \(y=\left(1+\cos^{2} 7x\right)^{3}\) is \(dy/dx = -42\left(1+\cos^{2}7x\right)^{2}\cos(7x)\sin(7x)\)

Step-by-step solution

Identify the outer and inner functions

Identify the inner and outer functions. Here, the inner function is \(1 + \cos^{2}(7x)\), and the outer function is \((u)^{3}\) where \(u\) is inner function.

Differentiate the outer function with respect to the inner function

The derivative of the outer function \(y=u^{3}\) with respect to the inner function \(u\) is \(3u^{2}\), according to the power rule of derivatives.

Differentiate the inner function

Differentiate the inner function, \(1 + \cos^{2}(7x)\). The derivative of constant 1 is 0. The derivative of \(\cos^{2}(7x)\) can be obtained using the chain rule. Differentiate \(\cos(7x)\) with respect to \(7x\) gives -\(\sin(7x)\). The derivative of \(7x\) with respect to \(x\) is \(7\). Therefore, the derivative of \(\cos^{2}(7x)\) is \(2\cos(7x)\) * \(-\sin(7x)\) * \(7\) = \(-14\cos(7x)\sin(7x)\)

Use the chain rule to get the final derivative

Apply the chain rule which is the derivative of the outer function times the derivative of the inner function. The chain rule gives, \(dy/dx = 3(1+\cos^{2}7x)^{2} * -14\cos(7x)\sin(7x)\)