Simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$, and evaluate as $h$ approaches 0 to find the rate of change of $f(x)$.
-$f(x)=\frac{1}{x^{4}}$

A) $\frac{5}{x^{4}}$

B) $-\frac{5}{x^{4}}$

C) $\frac{4}{x^{5}}$

D) $-\frac{4}{x^{5}}$

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 $f(x)$.
-$f(x)=\frac{1}{x^{4}}$

A) $\frac{5}{x^{4}}$

B) $-\frac{5}{x^{4}}$

C) $\frac{4}{x^{5}}$

D) $-\frac{4}{x^{5}}$

Quizplus · Accepted Answer

To simplify the difference quotient for $f(x) = \frac{1}{x^4}$, we start with $\frac{f(x+h)-f(x)}{h} = \frac{\frac{1}{(x+h)^4} - \frac{1}{x^4}}{h}$. Simplifying this expression involves finding a common denominator and then taking the limit as $h$ approaches 0. The result of this process is $-\frac{4}{x^5}$, which represents the rate of change of $f(x)$ at any point $x$. This is a direct application of the derivative of $f(x) = x^{-4}$, which is $f'(x) = -4x^{-5}$.

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