Differentiate y = (5x + 1)$^4$

To solve this problem, we need to apply the Chain Rule in calculus. The Chain Rule is used when we have a function nested inside another function, such as $y = [f(x)]^n$.

1. Analyze the Question

We are asked to find the derivative of $y = (5x + 1)^4$ with respect to $x$. This is written as $\frac{dy}{dx}$.

2. Explain the Concept

The Chain Rule states that if $y = u^n$, then:

In simpler terms:

1. Bring the power down to the front (multiply by the exponent).
2. Keep the expression inside the bracket exactly as it is and subtract 1 from the exponent.
3. Multiply by the derivative of the expression inside the bracket.

3. Show the Solution

Given: $y = (5x + 1)^4$

Step 1: Differentiate the outer function. Treat $(5x + 1)$ as a single unit ($u$). The derivative of $u^4$ is $4u^3$. So, we have: $4(5x + 1)^3$

Step 2: Differentiate the inner function. Now, differentiate the expression inside the bracket $(5x + 1)$ with respect to $x$. The derivative of $5x$ is $5$, and the derivative of $1$ is $0$. So, the derivative of $(5x + 1)$ is $5$.

Step 3: Combine the results. Multiply the result from Step 1 by the result from Step 2:

4. Identify the Answer

Comparing our result to the given options, we find that it matches option A.

5. Exam Tip

Whenever you see a bracket raised to a power, always remember the "Inside-Outside" rule: Differentiate the outside, keep the inside the same, then multiply by the derivative of the inside. Don't forget that last step—many students lose marks by forgetting to multiply by the derivative of the inner function!