Question

For a differentiable function $f$, the value of $Lt_{h\to0} \frac{\left[f\left(x+h\right)\right]^{2} - \left[f\left(x\right)\right]^{2}}{2h}$ is equal to

• A. $[f'(x)]^2$
• B. f(x) f'(x)
• C. $\frac{1}{2} [ f(x)]^2$
• D. $\frac{1}{2} [ f(x)]^2 - [f(x)]^2$

Answer 1

Answer: (B)

f(x) f'(x)

Answer 2

By def. the given limit $= \frac{1}{2} \frac{d}{dx} \left(f\left(x\right)\right)^{2} = \frac{1}{2} .2f\left(x\right).f'\left(x\right)$ $= f\left(x\right)f'\left(x\right)$