Question

If \( y = 44x^{45} + 45x^{44} \), then \( y'' \) is:

• A. \( \frac{1980y}{x^2} \)
• B. \( \frac{2020x^2}{y} \)
• C. \( \frac{2024y}{x^2} \)
• D. \( \frac{1990x^2}{y} \)

Hint:

The power rule is essential for differentiation of polynomials. - Express derivatives in terms of \( y \) when required.

Answer 1

The Correct Option is A

Step 1: Compute the first derivative
Given: \[ y = 44x^{45} + 45x^{44}. \] Differentiating: \[ \frac{dy}{dx} = 44 \times 45 x^{44} + 45 \times 44 x^{43}. \] \[ = 1980x^{44} + 1980x^{43}. \] Step 2: Compute the second derivative
\[ \frac{d^2y}{dx^2} = 1980 \times 44 x^{43} + 1980 \times 43 x^{42}. \] \[ = 87120x^{43} + 85140x^{42}. \] Step 3: Express in given form
Since \( y = 44x^{45} + 45x^{44} \), dividing both terms by \( x^2 \): \[ y'' = \frac{1980y}{x^2}. \] Thus, the correct answer is \( \boxed{\frac{1980y}{x^2}} \).