Question

If \(f(x)=(x^3+sin\pi x)^5\), then \(f'(1)\) is equal to

• A. 2⁵
• B. 5(2⁴)
• C. 15
• D. 5(3+π)
• E. 5(3-π)

Answer 1

Answer: (E)

We are given the function \( f(x) = (x^3 + \sin(\pi x))^5 \), and we need to find \( f'(1) \).

To solve this, we will use the chain rule for differentiation. Let: 

\( u = x^3 + \sin(\pi x) \), so that \( f(x) = u^5 \).

Now, we differentiate \( f(x) \) with respect to \( x \) using the chain rule:

\( f'(x) = 5u^4 \cdot \frac{du}{dx} \).

Next, we need to compute \( \frac{du}{dx} \):

\( u = x^3 + \sin(\pi x) \)

\( \frac{du}{dx} = 3x^2 + \pi \cos(\pi x) \).

Thus, the derivative of \( f(x) \) is:

\( f'(x) = 5(x^3 + \sin(\pi x))^4 \cdot (3x^2 + \pi \cos(\pi x)) \).

Now, we substitute \( x = 1 \) into this expression:

\( f'(1) = 5(1^3 + \sin(\pi \cdot 1))^4 \cdot (3 \cdot 1^2 + \pi \cos(\pi \cdot 1)) \).

Since \( \sin(\pi) = 0 \) and \( \cos(\pi) = -1 \), we have:

\( f'(1) = 5(1^3 + 0)^4 \cdot (3 \cdot 1^2 + \pi \cdot (-1)) \).

\( f'(1) = 5(1)^4 \cdot (3 - \pi) \).

\( f'(1) = 5(3 - \pi) \).

The correct answer is 5(3 - π).