Question

given ( y = f(u) ) and ( u = g(x) ), find ( \frac{dy}{dx}=f(g(x))g(x) ).
( y = 8u^{8},u = 4x + 19 )
( \frac{dy}{dx}=)

Explanation:

Step1: Find \(f^{\prime}(u)\)

Given \(y = f(u)=8u^{8}\), using the power rule \((x^{n})^\prime=nx^{n - 1}\), we have \(f^{\prime}(u)=\frac{d}{du}(8u^{8})=8\times8u^{7}=64u^{7}\).

Step2: Find \(g^{\prime}(x)\)

Given \(u = g(x)=4x + 19\), using the sum rule \((x^{n})^\prime=nx^{n - 1}\) (\((ax + b)^\prime=a\) for \(a,b\) constants), we have \(g^{\prime}(x)=\frac{d}{dx}(4x + 19)=4\).

Step3: Substitute into the chain - rule formula

By the chain - rule \(\frac{dy}{dx}=f^{\prime}(g(x))g^{\prime}(x)\). Substitute \(u = g(x)=4x + 19\) into \(f^{\prime}(u)\) and \(g^{\prime}(x) = 4\).
\(\frac{dy}{dx}=64(4x + 19)^{7}\times4\).

Step4: Simplify the expression

\(\frac{dy}{dx}=256(4x + 19)^{7}\).

Answer:

\(256(4x + 19)^{7}\)