If given a function $f(x) = x^{7} + 2x^{5} - 3x^2$, which method of differentiation would be most appropriate to find $f'(x)$?

Apply the Power Rule to each term separately.

Use the Quotient Rule on each term individually.

Apply the Chain Rule to the entire function as one unit.

Differentiate using implicit differentiation techniques.

Given that $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ exists for the function $f$, what can we infer about the behavior of $f$ around point 'a'?

There are no abrupt changes in the slope of the tangent to the curve of f near 'a'.

The tangent to f's curve has an undefined slope near 'a'.

There are potential cusps or corners on f's graph around point 'a'.

Discontinuities might exist in f’s graph close to ‘a’.

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2.5 Applying the Power Rule

3 min read•february 15, 2024

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Welcome back to AP Calculus with Fiveable! We are now diving into one of the most valuable fundamental concepts in calculus: the Power Rule. This is the first of many derivative rules that you’re going to learn about!

⚡ The Power Rule

The Power Rule states that if $f(x) = x^n$ , where $n$ is constant, then the derivative $f'(x)$ is given by:

f'(x) = n \cdot x^{(n-1)}

So, the Power Rule provides a shortcut for finding the derivative without using the limit definition of a derivative. Wasn’t doing that annoying?!

🏋️‍♂️ Practice Problems

Let’s work on a few questions to get the Power Rule down!

Given $f(x) = x^4$ , find $f'(x)$ .
Given $f(x) = \frac{1}{x^5}$ , find $f'(x)$ .
Given $f(x) = \sqrt{x}$ , find $f'(x)$ .
Given $f(x) = x^6 +2x^4-10$ , find $f'(x)$ .

💡 Before we reveal the answers, remember:

The Power Rule with fractions can be tricky! Sometimes rewriting the equation can help.
The derivative of any constant is zero.

👀 Answers to Practice Problems

Note how for many of these problems, the equations were rewritten before solving for the derivative using the power rule.

$f'(x) = 4 \cdot x^{(4-1)}$ $= 4x^3$
$f(x) = x^{-5}$

$f'(x) = -5 \cdot x^{(-5-1)}$ $= -5x^{-6}$ $= \frac{-5}{x^{6}}$
$f(x) = x^{1/2}$

$f'(x) = \frac{1}{2} \cdot x^\frac{-1}{2}$ $= \frac {1}{2\sqrt{x}}$
$f'(x)=6x^5 + 8x^3$

Yep! That’s it. This lesson was super short. Want to jump into the rest of the derivative rules you have to know? ⏭️

Key Terms to Review (6)

Differentiate

: Differentiation refers to finding the derivative of a function. It measures how quickly or slowly a quantity changes at any given point and provides information about its slope or rate of change.

f^-1(x) = √x

: The equation f^-1(x) = √x represents an inverse function that takes x as input and returns its square root as output. It is commonly used when finding inverses of quadratic functions.

Inverse Functions

: Inverse functions are two functions that "undo" each other. When you apply one function and then the inverse function, you get back to where you started.

nx^(n-1)

: The term "nx^(n-1)" represents the derivative of a power function, where n is a constant and x is the variable. It indicates how the rate of change of the function varies with respect to x.

Power Rule

: The power rule is a calculus rule used to find the derivative of a function that is raised to a constant power. It states that if f(x) = x^n, where n is a constant, then the derivative of f(x) with respect to x is equal to n*x^(n-1).

Product Rule

: The product rule is a formula used to find the derivative of the product of two functions. It states that the derivative of the product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function.