find the indicated derivative.\n$\frac{d}{du}(9u^{0.4}-8u^{3.7})$

Explanation:

Step1: Apply sum - difference rule

The derivative of a sum/difference of functions is the sum/difference of their derivatives. So, $\frac{d}{du}(9u^{0.4}-8u^{3.7})=\frac{d}{du}(9u^{0.4})-\frac{d}{du}(8u^{3.7})$.

Step2: Apply constant - multiple rule

The constant - multiple rule states that $\frac{d}{du}(cf(u)) = c\frac{d}{du}(f(u))$, where $c$ is a constant. So, $\frac{d}{du}(9u^{0.4}) = 9\frac{d}{du}(u^{0.4})$ and $\frac{d}{du}(8u^{3.7})=8\frac{d}{du}(u^{3.7})$.

Step3: Apply power rule

The power rule for differentiation is $\frac{d}{du}(u^n)=nu^{n - 1}$. For $n = 0.4$, $9\frac{d}{du}(u^{0.4})=9\times0.4u^{0.4 - 1}=3.6u^{- 0.6}$. For $n = 3.7$, $8\frac{d}{du}(u^{3.7})=8\times3.7u^{3.7 - 1}=29.6u^{2.7}$.

Step4: Combine results

$\frac{d}{du}(9u^{0.4}-8u^{3.7})=3.6u^{-0.6}-29.6u^{2.7}=\frac{3.6}{u^{0.6}}-29.6u^{2.7}$.

Answer:

$\frac{3.6}{u^{0.6}}-29.6u^{2.7}$