Simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$, and evaluate as $h$ approaches 0 to find the rate of change of $f(x)$.
-$f(x)=\sqrt{5 x}$

A) $\frac{5}{2 \sqrt{5 x}}$

B) $-\frac{2}{\sqrt{5 x}}$

C) $-\frac{5}{2 \sqrt{5 x}}$

D) $\frac{1}{\sqrt{5 x}}$

Question

Simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$, and evaluate as $h$ approaches 0 to find the rate of change of $f(x)$.
-$f(x)=\sqrt{5 x}$

A) $\frac{5}{2 \sqrt{5 x}}$

B) $-\frac{2}{\sqrt{5 x}}$

C) $-\frac{5}{2 \sqrt{5 x}}$

D) $\frac{1}{\sqrt{5 x}}$

Quizplus · Accepted Answer

The difference quotient for $f(x) = \sqrt{5x}$ simplifies to $\frac{\sqrt{5(x+h)} - \sqrt{5x}}{h}$. Multiplying the numerator and denominator by the conjugate of the numerator, $\sqrt{5(x+h)} + \sqrt{5x}$, simplifies to $\frac{5}{2\sqrt{5x}}$ as $h$ approaches 0. This represents the rate of change of $f(x)$ at any point $x$.

Simplify the Difference Quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ , and Evaluate As hhh Approaches 0 to Find the Rate of Change Of

Simplify the Difference Quotient $\frac{f(x+h)-f(x)}{h}$ , and Evaluate As $h$ Approaches 0 to Find the Rate of Change Of