Question

differentiate the function.
g(t) = ln(7 + t^{8})
g(t) = \frac{56t^{7}+t^{8}}{7 + t^{8}}

Explanation:

Step1: Apply the chain rule

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 \(g(t)=\ln(u)\) where \(u = 7 + t^{8}\), the derivative of \(\ln(u)\) with respect to \(u\) is \(\frac{1}{u}\), and the derivative of \(u = 7 + t^{8}\) with respect to \(t\) is \(8t^{7}\).

Step2: Combine the derivatives

By the chain rule, \(g^{\prime}(t)=\frac{1}{7 + t^{8}}\cdot(8t^{7})\).

Answer:

\(\frac{8t^{7}}{7 + t^{8}}\)