Question

differentiate the function.
y = \frac{1}{(5x - 3)^4}
\frac{dy}{dx} = \square

Explanation:

Step1: Rewrite the function

Rewrite \( y = \frac{1}{(5x - 3)^4} \) as \( y=(5x - 3)^{-4} \) using the negative exponent rule \( \frac{1}{a^n}=a^{-n} \).

Step2: Apply the chain rule

The chain rule states that if \( y = u^n \) and \( u = f(x) \), then \( \frac{dy}{dx}=n u^{n - 1}\cdot\frac{du}{dx} \). Let \( u = 5x-3 \), so \( n=-4 \), \( \frac{du}{dx}=5 \).
First, find the derivative of \( y \) with respect to \( u \): \( \frac{dy}{du}=-4u^{-5} \).
Then, multiply by \( \frac{du}{dx} \): \( \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=-4u^{-5}\cdot5 \).

Step3: Substitute back \( u = 5x - 3 \)

Substitute \( u = 5x - 3 \) into the expression: \( \frac{dy}{dx}=-20(5x - 3)^{-5} \).
Rewrite using positive exponents: \( \frac{dy}{dx}=-\frac{20}{(5x - 3)^5} \).

Answer:

\( -\dfrac{20}{(5x - 3)^5} \)