Question

hw14 the chain rule (target c4; §3.6)
due: thu oct 9, 2025 11:59pm
attempt 1
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hw14 the chain rule (target c4; §3.6)
score: 3/11 answered: 3/11
question 4
use the chain rule to find the derivative of (f(x)=5e^{6x^{8}-7x^{9}})
(f(x)=)

Explanation:

Step1: Recall chain - rule formula

The chain - rule states that if \(y = f(u)\) and \(u = g(x)\), then \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\). For \(y = 5e^{u}\) where \(u = 6x^{8}-7x^{9}\), first find \(\frac{dy}{du}\) and \(\frac{du}{dx}\).

Step2: Differentiate \(y = 5e^{u}\) with respect to \(u\)

The derivative of \(y = 5e^{u}\) with respect to \(u\) is \(\frac{dy}{du}=5e^{u}\).

Step3: Differentiate \(u = 6x^{8}-7x^{9}\) with respect to \(x\)

Using the power - rule \(\frac{d}{dx}(ax^{n})=nax^{n - 1}\), we have \(\frac{du}{dx}=(6\times8x^{7})-(7\times9x^{8}) = 48x^{7}-63x^{8}\).

Step4: Apply the chain - rule

By the chain - rule \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\), substitute \(u = 6x^{8}-7x^{9}\), \(\frac{dy}{du}=5e^{u}\) and \(\frac{du}{dx}=48x^{7}-63x^{8}\) into the formula. So \(f^{\prime}(x)=5e^{6x^{8}-7x^{9}}(48x^{7}-63x^{8})\).

Answer:

\(5(48x^{7}-63x^{8})e^{6x^{8}-7x^{9}}\)