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