Question

h(x) = (5 - 6x)^5

h(x) = ?

Explanation:

Step1: Let \(u = 5 - 6x\)

Then \(h(x)=u^{5}\)

Step2: Find the derivative of \(u\) with respect to \(x\)

\(u^\prime=\frac{d}{dx}(5 - 6x)=-6\)

Step3: Find the derivative of \(h(u)\) with respect to \(u\)

Using the power rule \(\frac{d}{du}(u^{n})=nu^{n - 1}\), for \(n = 5\), we have \(h^\prime(u)=\frac{d}{du}(u^{5})=5u^{4}\)

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

Substitute \(u = 5 - 6x\), \(h^\prime(u)=5u^{4}\) and \(u^\prime=-6\) into the chain - rule formula:
\(h^\prime(x)=5(5 - 6x)^{4}\cdot(-6)\)

Answer:

\(h^\prime(x)=-30(5 - 6x)^{4}\)