Applying the Power Rule - AP Calculus BC Study Guide

The Core Idea: Applying the Power Rule

The fundamental goal of this topic is to learn a highly efficient method for finding the derivative of a specific, yet common, type of function: a variable raised to an integer power. Before this, derivatives were often found using the cumbersome limit definition. The power rule provides a direct computational shortcut for functions of the form $f(x)=x^n$, where $n$ is any integer (positive, negative, or zero).

This rule is a foundational building block in differential calculus. Mastering its application is essential for differentiating more complex functions, such as polynomials, which are sums of these power functions. The power rule allows us to move from the conceptual definition of the derivative to a practical, procedural method for calculating it.

Key Formulas

The primary and sole formula for this topic is the Power Rule for differentiation.

The Power Rule

For a function of the form $f(x)=x^n$, where $n$ is any integer, its derivative with respect to $x$ is given by:

$$\frac{d}{dx}(x^n)=n{x}^{n-1}$$

This formula states that to find the derivative, you multiply the function by its original exponent ($n$) and then decrease the original exponent by one ($n-1$).

Understanding the Rule

The power rule is a two-step process applied to functions of the form $f(x)=x^n$. It is crucial to recognize that the base must be the variable ($x$) and the exponent must be a constant integer ($n$).

1. Multiply by the Exponent: The original exponent, $n$, becomes the coefficient of the new expression.

2. Subtract One from the Exponent: The new exponent is the original exponent minus one, or $n-1$.

This rule works for all integer exponents:

• Positive Integers: For $f(x)=x^3$, $n=3$. The derivative is $f^{\prime }(x)=3x^{3-1}=3x^2$.

• Negative Integers: For $g(x)=x^{-4}$, $n=-4$. The derivative is $g^{\prime }(x)=-4x^{-4-1}=-4x^{-5}$.

• Zero: For $h(x)=x^0$ (which is equivalent to $h(x)=1$ for $x\ne 0$), $n=0$. The derivative is $h^{\prime }(x)=0x^{0-1}=0x^{-1}=0$. This is consistent with the rule that the derivative of a constant is zero.

Functions must often be rewritten algebraically to fit the $x^n$ form before the rule can be applied. For example, $f(x)=\frac{1}{x^5}$ must first be rewritten as $f(x)=x^{-5}$.

Core Concepts & Rules

• Derivative of Power Functions: The power rule is the specific method for finding the derivative of functions in the form $f(x)=x^n$.

• The Rule: The derivative of $x^n$ is $n{x}^{n-1}$.

• Integer Exponents: The power rule, as defined in this topic, applies specifically when the exponent $n$ is an integer (e.g., $...-3,-2,-1,0,1,2,\mathrm{3...}$).

• Algebraic Manipulation: Functions may need to be rewritten into the standard $x^n$ form before the power rule can be applied. For instance, a variable in the denominator corresponds to a negative exponent.

Step-by-Step Example 1: Basic Application

Problem: Find the derivative of the function $f(x)=x^7$.

Solution:

• Step 1: Identify the form of the function.

  The function $f(x)=x^7$ is in the form $f(x)=x^n$. We can identify the integer exponent as $n=7$.

• Step 2: Apply the Power Rule formula.

  The power rule states that $\frac{d}{dx}(x^n)=n{x}^{n-1}$.

• Step 3: Substitute the value of $n$ into the formula.

  Substitute $n=7$ into the rule:

  $$f^{\prime }(x)=7x^{7-1}$$

• Step 4: Simplify the expression.

  Subtract 1 from the exponent:

  $$f^{\prime }(x)=7x^6$$

Final Answer: The derivative of $f(x)=x^7$ is $f^{\prime }(x)=7x^6$.

Step-by-Step Example 2: Exam-Style Application

Problem: Find the slope of the tangent line to the graph of $g(x)=\frac{1}{x^4}$ at the point where $x=2$.

Solution:

• Step 1: Rewrite the function in $x^n$ form.

  The function is given as a fraction. To apply the power rule, we must express it with an exponent. Using the rules of exponents, $\frac{1}{a^m}=a^{-m}$.

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

• Step 2: Differentiate using the Power Rule.

  Now the function is in the form $g(x)=x^n$, where $n=-4$. We apply the power rule $\frac{d}{dx}(x^n)=n{x}^{n-1}$.

  $$g^{\prime }(x)=-4x^{-4-1}$$

• Step 3: Simplify the derivative.

  Be careful with the negative exponent. $-4-1=-5$.

  $$g^{\prime }(x)=-4x^{-5}$$

  It is often helpful to rewrite the derivative with a positive exponent for evaluation:

  $$g^{\prime }(x)=-\frac{4}{x^5}$$

• Step 4: Evaluate the derivative at the given point.

  The slope of the tangent line at a specific point is the value of the derivative at that point. We need to find $g^{\prime }(2)$.

  $$g^{\prime }(2)=-\frac{4}{(2{)}^5}$$

• Step 5: Calculate the final value.

  $$g^{\prime }(2)=-\frac{4}{32}=-\frac{1}{8}$$

Final Answer: The slope of the tangent line to $g(x)=\frac{1}{x^4}$ at $x=2$ is $-\frac{1}{8}$.

Using Your Calculator

This topic covers a fundamental analytical skill. The power rule is a process you must be able to perform by hand, without a calculator. A calculator is not used to find the derivative using the power rule, but it is an excellent tool for verifying your answer.

You can check the result of a derivative at a specific point using your calculator's numerical differentiation feature.

Example: Verify the answer to Example 2 ($g^{\prime }(2)$ for $g(x)=\frac{1}{x^4}$).

1. Access the numerical derivative function on your calculator. On a TI-84 style calculator, this is often found by pressing [MATH] and selecting nDeriv( or by using the shortcut $[alpha][window]$.

2. Enter the expression in the required format. The syntax is typically $d/dx(function)|x=value$.

3. Input the original function and the point of evaluation:

   $$\frac{d}{dx}\left(\frac{1}{x^4}\right){|}_{x=2}$$

4. The calculator will return a numerical approximation. In this case, it will return $-0.125$.

5. Compare this decimal to your analytical answer. Since $-\frac{1}{8}=-0.125$, your hand-calculated answer is confirmed to be correct.

Note: The calculator computes a numerical approximation of the derivative. Your answer should always be the exact, analytical result found using the power rule.

AP Exam Quick Hit

Common Question Types

• Direct Differentiation: You will be asked to find the derivative of a simple power function. This skill is most often a single step within a larger problem.

  • Example: "If $y=x^{-10}$, find $\frac{dy}{dx}$."

• Finding the Slope or Equation of a Tangent Line: You will be asked to apply the power rule and then use the result to find the slope of the tangent line at a given x-value.

  • Example: "What is the slope of the line tangent to the curve $f(x)=x^4$ at $x=-1$?"

• Functions Requiring Algebraic Rewriting: Questions will present functions where the variable is in the denominator, requiring you to first rewrite the function with a negative integer exponent before differentiating.

  • Example: "Find $f^{\prime }(x)$ if $f(x)=\frac{1}{x}$."

Common Mistakes

• Forgetting to Subtract One from the Exponent: A frequent error is to bring the exponent down as a coefficient but forget to change the exponent. For $f(x)=x^5$, a common incorrect answer is $5x^5$ instead of the correct $5x^4$.

• Incorrectly Handling Negative Exponents: When differentiating a term like $x^{-3}$, students often make an arithmetic error with the new exponent. They might incorrectly write $-3x^{-2}$ (by adding 1) instead of the correct $-3x^{-4}$ (by subtracting 1, since $-3-1=-4$).

• Misapplying the Power Rule: Students may try to apply the power rule to functions where it is not applicable. The rule $\frac{d}{dx}(x^n)=n{x}^{n-1}$ is for a variable base and a constant exponent. It cannot be used for an exponential function like $f(x)=2^x$, which has a constant base and a variable exponent.

• Algebraic Errors: Before differentiation can even begin, students might make a mistake rewriting a function. For example, incorrectly rewriting $\frac{1}{x^3}$ as $x^{1/3}$ or $x^3$ instead of the correct $x^{-3}$.