Question

If $f(x)=x|x|$, then $f^{\prime}(-10)=$ ________.

• A. -20
• B. -10
• C. -40
• D. 20
• E. 40

Hint:

The derivative of $x|x|$ is $2|x|$.

Answer 1

The Correct Option is D

To find the derivative of the function \( f(x) = x|x| \) at a specific point \( x = -10 \), we can start by understanding the behavior of \( f(x) \). The function \( f(x) = x|x| \) is a piecewise function. The representation of \( f(x) \) depends on the sign of \( x \):

• If \( x \geq 0 \), then \( |x| = x \). Thus, \( f(x) = x \cdot x = x^2 \).
• If \( x < 0 \), then \( |x| = -x \). Thus, \( f(x) = x \cdot (-x) = -x^2 \).

For \( x = -10 \), since \( x < 0 \), we have:

\(f(x) = -x^2\)

Now, we need to find the derivative of \( f(x) = -x^2 \):

\(f^{\prime}(x) = \frac{d}{dx}(-x^2) = -2x\)

Substitute \( x = -10 \) into the derivative:

\(f^{\prime}(-10) = -2(-10) = 20\)

This matches the correct answer.

Thus, the derivative of \( f(x) \) at \( x = -10 \) is 20, which is the correct answer choice. Therefore, the answer is:

20