Question

calculate the derivative of the following function.
y=(4x + 7)^{\frac{1}{5}}
\frac{dy}{dx}=\square

Explanation:

Step1: Identify the outer - inner functions

Let $u = 4x + 7$, then $y = u^{\frac{1}{5}}$.

Step2: Differentiate the outer function with respect to u

Using the power rule $\frac{d}{du}(u^n)=nu^{n - 1}$, for $y = u^{\frac{1}{5}}$, we have $\frac{dy}{du}=\frac{1}{5}u^{-\frac{4}{5}}$.

Step3: Differentiate the inner function with respect to x

For $u = 4x+7$, $\frac{du}{dx}=4$.

Step4: Apply the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=\frac{1}{5}u^{-\frac{4}{5}}$ and $\frac{du}{dx}=4$ into the chain - rule formula, and replace $u$ with $4x + 7$. So $\frac{dy}{dx}=\frac{1}{5}(4x + 7)^{-\frac{4}{5}}\cdot4=\frac{4}{5}(4x + 7)^{-\frac{4}{5}}$.

Answer:

$\frac{4}{5}(4x + 7)^{-\frac{4}{5}}$