Applying the Power Rule - AP Calculus AB Study Guide

The Core Idea: Applying the Power Rule

The Power Rule is a fundamental shortcut for finding the derivative of a function. Instead of using the cumbersome limit definition of the derivative, the Power Rule provides a direct and efficient method for differentiating functions of the form $f(x)=x^n$, where $n$ is any real number. This rule is the cornerstone of differentiation for a large class of functions, including polynomials.

When combined with the Constant Multiple Rule and the Sum/Difference Rule, the Power Rule allows us to differentiate any polynomial function term-by-term. Furthermore, by using algebraic manipulation to rewrite functions involving radicals (like $\sqrt{x}$) or reciprocals (like $\frac{1}{x^2}$) into the form $x^n$, we can extend the application of the Power Rule to a much broader set of functions. Mastering this topic is about both applying the rule itself and recognizing when a function must first be rewritten into the proper form.

Key Rules

The process of differentiating polynomials and other power functions relies on three foundational rules.

1. The Power Rule

   For any real number $n$, the derivative of the function $f(x)=x^n$ is given by:

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

   This means you bring the original exponent down as a coefficient and then subtract one from the original exponent to get the new exponent.

2. The Constant Multiple Rule

   The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

   $$\frac{d}{dx}[c\cdot f(x)]=c\cdot f^{\prime }(x)$$

   This rule allows us to "pull out" constant coefficients before differentiating the variable part of a term.

3. The Sum and Difference Rule

   The derivative of a sum or difference of functions is the sum or difference of their individual derivatives.

   $$\frac{d}{dx}[f(x)\pm g(x)]=f^{\prime }(x)\pm g^{\prime }(x)$$

   This rule allows us to differentiate complex functions one term at a time.

Understanding Rewriting Functions

The Power Rule can only be applied directly to expressions in the form $x^n$. A critical skill for success in differentiation is the ability to use algebra and exponent rules to rewrite functions into this form before applying the derivative rules. Many functions that do not initially appear to be power functions can be differentiated using the Power Rule after a preliminary algebraic step.

• Radical Functions: Convert roots to rational exponents.

  • $\sqrt{x}$ must be rewritten as $x^{1/2}$.

  • $\sqrt[3]{x^2}$ must be rewritten as $x^{2/3}$.

• Rational Functions with Monomial Denominators: Convert expressions with variables in the denominator to negative exponents.

  • $\frac{1}{x^3}$ must be rewritten as $x^{-3}$.

  • $\frac{5}{x^4}$ must be rewritten as $5x^{-4}$.

• Expressions Requiring Simplification: Sometimes, you must simplify an expression by distributing or dividing before you can differentiate.

  • For $f(x)=\frac{x^3+2x}{x}$, you must first simplify to $f(x)=x^2+2$ before differentiating.

Failing to rewrite the function into the proper form is a common source of error. Always check if a function can be simplified or rewritten before you begin to take the derivative.

Core Concepts & Rules

• The Power Rule is a shortcut: It replaces the need for the limit definition of the derivative for functions of the form $f(x)=x^n$.

• The exponent $n$ can be any real number: This includes positive integers, negative integers, fractions, and even irrational numbers.

• Differentiate term-by-term: The Sum and Difference Rule allows you to break down a polynomial into individual terms and find the derivative of each one separately.

• Constants "come along for the ride": The Constant Multiple Rule means that coefficients are not changed by the process of differentiation, they are simply multiplied by the derivative of the variable part.

• The derivative of a constant is zero: For a function $f(x)=c$, where $c$ is a constant, $f^{\prime }(x)=0$. This can be thought of as a special case of the power rule: $f(x)=c\cdot x^0$, so $f^{\prime }(x)=c\cdot 0\cdot x^{-1}=0$.

• Rewrite before you differentiate: You must convert radicals and variables in the denominator into the form $a{x}^n$ before applying the Power Rule.

Step-by-Step Example 1: Basic Application

Problem: Find the derivative of the function $f(x)=3x^4-5x^2+7x-2$.

Solution:

1. Apply the Sum/Difference Rule: We can differentiate the function term by term.

   $$f^{\prime }(x)=\frac{d}{dx}(3x^4)-\frac{d}{dx}(5x^2)+\frac{d}{dx}(7x)-\frac{d}{dx}(2)$$

2. Differentiate the first term, $3x^4$: Use the Constant Multiple Rule and the Power Rule.

   $$\frac{d}{dx}(3x^4)=3\cdot \frac{d}{dx}(x^4)=3\cdot (4x^{4-1})=12x^3$$

3. Differentiate the second term, $-5x^2$:

   $$\frac{d}{dx}(-5x^2)=-5\cdot \frac{d}{dx}(x^2)=-5\cdot (2x^{2-1})=-10x^1=-10x$$

4. Differentiate the third term, $7x$: Remember that $x$ is $x^1$.

   $$\frac{d}{dx}(7x)=7\cdot \frac{d}{dx}(x^1)=7\cdot (1x^{1-1})=7\cdot (1x^0)=7\cdot 1=7$$

5. Differentiate the constant term, $-2$: The derivative of any constant is zero.

   $$\frac{d}{dx}(-2)=0$$

6. Combine the results: Put all the differentiated terms back together.

   $$f^{\prime }(x)=12x^3-10x+7-0$$

   $$f^{\prime }(x)=12x^3-10x+7$$

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

Problem: Find the equation of the line tangent to the graph of $g(x)=\frac{4x^3+\sqrt{x}}{x}$ at the point where $x=1$.

Solution:

1. Rewrite the function: The function is not in a form where we can apply the Power Rule directly. We must first simplify by dividing each term in the numerator by the denominator, $x$.

   $$g(x)=\frac{4x^3}{x}+\frac{\sqrt{x}}{x}$$

   $$g(x)=4x^2+\frac{x^{1/2}}{x^1}$$

   Using the rule of exponents $\frac{x^a}{x^b}=x^{a-b}$:

   $$g(x)=4x^2+x^{1/2-1}=4x^2+x^{-1/2}$$

   Now the function is ready to be differentiated.

2. Find the derivative, $g^{\prime }(x)$: Apply the Power Rule to each term.

   $$g^{\prime }(x)=\frac{d}{dx}(4x^2)+\frac{d}{dx}(x^{-1/2})$$

   $$g^{\prime }(x)=4\cdot (2x^{2-1})+(-\frac{1}{2}{x}^{-1/2-1})$$

   $$g^{\prime }(x)=8x-\frac{1}{2}{x}^{-3/2}$$

3. Find the slope of the tangent line: The slope is the value of the derivative at $x=1$. Evaluate $g^{\prime }(1)$.

   $$m=g^{\prime }(1)=8(1)-\frac{1}{2}(1{)}^{-3/2}$$

   Since $1$ to any power is $1$:

   $$m=8-\frac{1}{2}(1)=7.5=\frac{15}{2}$$

4. Find the point of tangency: We have the x-coordinate, $x=1$. We need the y-coordinate by evaluating the original function $g(1)$.

   $$y=g(1)=4(1{)}^2+(1{)}^{-1/2}=4+1=5$$

   The point of tangency is $(1,5)$.

5. Write the equation of the tangent line: Use the point-slope form, $y-y_1=m(x-x_1)$.

   $$y-5=\frac{15}{2}(x-1)$$

Using Your Calculator

The Power Rule is a purely analytical technique, meaning it is performed by hand using algebraic rules. A calculator cannot apply the Power Rule for you to find a general derivative function like $f^{\prime }(x)=2x$.

However, a graphing calculator is an excellent tool for checking your answer at a specific point. The numerical derivative function (often nDeriv or found in the MATH menu as $d/dx($) can calculate the value of the derivative at a single point.

To check the result of Example 2:

We found that $g^{\prime }(x)=8x-\frac{1}{2}{x}^{-3/2}$ and that $g^{\prime }(1)=7.5$.

1. On your calculator's home screen, use the numerical derivative command. The syntax is typically: nDeriv(function, variable, value).

2. Enter the original function, $g(x)$. Be careful with parentheses.

   • nDeriv((4x^3 + sqrt(x))/x, x, 1)

3. The calculator will return $7.5$, confirming that our derivative calculation is correct at that specific point.

This method does not give you the derivative function $g^{\prime }(x)$, but it provides a reliable way to verify your work for problems that ask for the slope or derivative at a single x-value.

AP Exam Quick Hit

Common Question Types

• Direct Derivative Calculation: You will be given a function involving polynomials, radicals, or negative exponents and asked to find its derivative.

  • Example: "If $f(x)=5x^3-\frac{2}{x^2}+\sqrt[3]{x}$, what is $f^{\prime }(x)$?"

• Finding the Equation of a Tangent Line: This multi-step problem requires you to find the derivative, evaluate it at a point to find the slope, find the y-coordinate at that point, and write the equation of the line.

  • Example: "Find the equation of the line tangent to the graph of $y=x(x^2-3)$ at $x=-1$."

• Finding Horizontal Tangents: This involves finding the derivative, setting it equal to zero, and solving for the x-values where the slope is zero.

  • Example: "Find all x-values for which the function $f(x)=2x^3+3x^2-12x+1$ has a horizontal tangent."

Common Mistakes

• Algebraic Rewriting Errors: The most frequent mistakes happen before any calculus is done. Forgetting to convert $\sqrt{x}$ to $x^{1/2}$ or incorrectly rewriting $\frac{3}{x^4}$ as $3x^4$ instead of $3x^{-4}$.

• Errors with Negative Exponents: When applying the power rule to a negative exponent, remember that subtracting 1 makes the exponent more negative. For example, the derivative of $x^{-4}$ is $-4x^{-5}$, not $-4x^{-3}$.

• Forgetting the Derivative of a Constant is Zero: Students sometimes mistakenly carry a constant term into the derivative. The derivative of $f(x)=x^2+5$ is $f^{\prime }(x)=2x$, not $2x+5$.

• Misapplying the Power Rule: The Power Rule only works for functions of the form $x^n$. Do not attempt to apply it to other types of functions, such as exponential functions (e.g., $2^x$) or trigonometric functions (e.g., $\sin (x)$). These have their own distinct derivative rules.

• Distributing a Negative Sign Incorrectly: When differentiating a function like $f(x)=5x^2-(x^3+2x)$, be sure to distribute the negative sign to all terms inside the parentheses before differentiating, or differentiate inside the parentheses first and then distribute the negative.