Question

calculate the derivative of the following function.
y = (4x - 19)^5
\frac{dy}{dx}=square

Explanation:

Step1: Apply chain - rule

Let $u = 4x - 19$, then $y = u^{5}$. The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$.

Step2: Find $\frac{dy}{du}$

Differentiate $y = u^{5}$ with respect to $u$. Using the power rule $\frac{d}{du}(u^{n})=nu^{n - 1}$, we get $\frac{dy}{du}=5u^{4}$.

Step3: Find $\frac{du}{dx}$

Differentiate $u = 4x - 19$ with respect to $x$. $\frac{du}{dx}=4$.

Step4: Calculate $\frac{dy}{dx}$

Substitute $\frac{dy}{du}$ and $\frac{du}{dx}$ into the chain - rule formula: $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=5u^{4}\cdot4$. Replace $u$ with $4x - 19$, so $\frac{dy}{dx}=20(4x - 19)^{4}$.

Answer:

$20(4x - 19)^{4}$