Question

If \( y = ax^r - bx^{r-1} \), then \[ \frac{d^2y}{dx^2} = ? \]

• A. \( r \cdot (r - 1) \cdot a x^{r - 2} - (r - 1) \cdot b \cdot x^{r - 2} \)
• B. \( r \cdot (r - 1) \cdot a x^{r - 2} - r \cdot b \cdot x^{r - 2} \)
• C. \( r^2 a x^{r - 2} - (r - 1) \cdot b x^{r - 2} \)
• D. \( r^2 a x^{r - 2} - r \cdot b x^{r - 2} \)

Hint:

When taking derivatives of power functions, apply the power rule and simplify step by step to avoid mistakes in the calculations.

Answer 1

The Correct Option is D

Step 1: Finding the first derivative.
We are given the function:
\[ y = ax^r - bx^{r-1}. \]
We need to find \( \frac{d^2y}{dx^2} \), the second derivative of \( y \). First, we find the first derivative \( \frac{dy}{dx} \) by applying the power rule:
\[ \frac{dy}{dx} = a \cdot r \cdot x^{r-1} - b \cdot (r-1) \cdot x^{r-2}. \]

Step 2: Finding the second derivative.
Now, differentiate \( \frac{dy}{dx} \) to find the second derivative:
\[ \frac{d^2y}{dx^2} = a \cdot r \cdot (r - 1) \cdot x^{r-2} - b \cdot (r - 1) \cdot x^{r-2}. \]

Step 3: Simplifying the second derivative.
Factor out \( (r - 1) x^{r-2} \):
\[ \frac{d^2y}{dx^2} = (r - 1) x^{r-2} \left( a \cdot r - b \right). \]
Final Answer:
Thus, the second derivative is:
\[ \boxed{r^2 a x^{r - 2} - r \cdot b x^{r - 2}}. \]