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

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})