Derivatives of $\\cos x$, $\\sin x$, $e^x$, and $\\ln x$ - AP Calculus AB Study Guide

The Core Idea: Derivatives of $\cos x$, $\sin x$, $e^x$, and $\ln x$

Calculus provides a powerful set of tools for analyzing the rate of change of functions. While the Power Rule allows us to differentiate polynomial functions, many important mathematical models in science, engineering, and economics rely on trigonometric and transcendental functions. This topic introduces the four essential, non-algebraic derivative rules that form the foundation for differentiating these functions: the derivatives of sine, cosine, the natural exponential function ($e^x$), and the natural logarithm function ($\ln x$).

Mastering these four specific rules is not just about memorization; it's about expanding our toolkit to analyze a much broader class of functions. These rules, often used in combination with the Product, Quotient, and Chain Rules, allow us to find slopes of tangent lines, solve optimization problems, and analyze the motion of objects whose behavior is described by these fundamental functions.

Key Formulas

These four derivative formulas are foundational and must be memorized. They are derived from the limit definition of the derivative and are presented here as established rules.

• Derivative of Sine

  $$\frac{d}{dx}(\sin x)=\cos x$$

• Derivative of Cosine

  $$\frac{d}{dx}(\cos x)=-\sin x$$

• Derivative of the Natural Exponential Function

  $$\frac{d}{dx}(e^x)=e^x$$

• Derivative of the Natural Logarithm Function

  $$\frac{d}{dx}(\ln x)=\frac{1}{x},\text{for }x>0$$

Understanding the Application

The four derivative rules presented in this topic are fundamental building blocks. On the AP Exam, it is rare for these rules to be tested in complete isolation. The primary challenge and application of this topic is to correctly use these rules in conjunction with other differentiation rules you have already learned, such as the Constant Multiple, Sum/Difference, Product, and Quotient Rules.

For example, to differentiate a function like $f(x)=x^2\sin x$, you must recognize it as a product of two functions, $x^2$ and $\sin x$. You would then apply the Product Rule, using the Power Rule for $x^2$ and the new rule for $\sin x$. Similarly, a function like $g(x)=\frac{e^x}{x}$ would require the Quotient Rule.

The most important combination is with the Chain Rule. While the rules are given for $\sin x$, $\cos x$, $e^x$, and $\ln x$, you will frequently encounter compositions like $\sin (5x)$, $e^{x^2}$, or $\ln (x^3+1)$. In these cases, you must apply the Chain Rule by taking the derivative of the "outside" function (using the rules from this topic) and multiplying by the derivative of the "inside" function. The ability to identify the correct rule or combination of rules is the key skill.

Core Concepts & Rules

• The derivative of the sine function is the cosine function: $\frac{d}{dx}(\sin x)=\cos x$.

• The derivative of the cosine function is the negative sine function: $\frac{d}{dx}(\cos x)=-\sin x$.

• The natural exponential function, $f(x)=e^x$, is unique in that it is its own derivative: $\frac{d}{dx}(e^x)=e^x$.

• The derivative of the natural logarithm function, $f(x)=\ln x$, is its reciprocal function: $\frac{d}{dx}(\ln x)=\frac{1}{x}$.

• These four rules must be committed to memory.

• These rules are applied term-by-term when functions are added or subtracted.

• For functions that involve products, quotients, or compositions, these rules must be used as part of the Product Rule, Quotient Rule, or Chain Rule, respectively.

Step-by-Step Example 1: Basic Application

Problem: Find the derivative of the function $f(x)=4\cos x-9e^x+3\ln x$.

Solution:

The function is a sum and difference of terms, so we can differentiate it term-by-term using the Sum and Difference Rules and the Constant Multiple Rule.

Step 1: Identify the individual terms to differentiate.

The terms are $4\cos x$, $-9e^x$, and $3\ln x$.

Step 2: Differentiate the first term, $4\cos x$.

Using the Constant Multiple Rule and the rule for the derivative of cosine:

$$\frac{d}{dx}(4\cos x)=4\cdot \frac{d}{dx}(\cos x)=4(-\sin x)=-4\sin x$$

Step 3: Differentiate the second term, $-9e^x$.

Using the Constant Multiple Rule and the rule for the derivative of $e^x$:

$$\frac{d}{dx}(-9e^x)=-9\cdot \frac{d}{dx}(e^x)=-9(e^x)=-9e^x$$

Step 4: Differentiate the third term, $3\ln x$.

Using the Constant Multiple Rule and the rule for the derivative of $\ln x$:

$$\frac{d}{dx}(3\ln x)=3\cdot \frac{d}{dx}(\ln x)=3\left(\frac{1}{x}\right)=\frac{3}{x}$$

Step 5: Combine the derivatives of each term to find the final answer.

$$f^{\prime }(x)=-4\sin x-9e^x+\frac{3}{x}$$

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

Problem: Find the equation of the line normal to the graph of $g(x)=e^x\sin x$ at $x=0$.

Solution:

A normal line is perpendicular to the tangent line. To find its equation, we first need the slope of the tangent line at $x=0$, which is the value of the derivative $g^{\prime }(0)$. The slope of the normal line will be the negative reciprocal of the tangent slope.

Step 1: Find the point of tangency on the graph.

We evaluate the function at $x=0$:

$$g(0)=e^0\sin (0)=1\cdot 0=0$$

The point is $(0,0)$.

Step 2: Find the derivative function, $g^{\prime }(x)$.

The function $g(x)$ is a product of two functions, $u(x)=e^x$ and $v(x)=\sin x$. We must use the Product Rule: $g^{\prime }(x)=u^{\prime }(x)v(x)+u(x)v^{\prime }(x)$.

• $u(x)=e^x⟹u^{\prime }(x)=e^x$

• $v(x)=\sin x⟹v^{\prime }(x)=\cos x$

Applying the Product Rule:

$$g^{\prime }(x)=(e^x)(\sin x)+(e^x)(\cos x)$$

Step 3: Calculate the slope of the tangent line at $x=0$.

Evaluate $g^{\prime }(x)$ at $x=0$:

$$g^{\prime }(0)=e^0\sin (0)+e^0\cos (0)=(1)(0)+(1)(1)=1$$

The slope of the tangent line, $m_{tan}$, is 1.

Step 4: Calculate the slope of the normal line.

The slope of the normal line, $m_{normal}$, is the negative reciprocal of the tangent slope.

$$m_{normal}=-\frac{1}{m_{tan}}=-\frac{1}{1}=-1$$

Step 5: Write the equation of the normal line.

Using the point-slope form, $y-y_1=m(x-x_1)$, with the point $(0,0)$ and slope $m=-1$:

$$y-0=-1(x-0)$$

$$y=-x$$

Using Your Calculator

The derivative rules for $\cos x$, $\sin x$, $e^x$, and $\ln x$ are analytical rules that must be memorized and applied by hand on the no-calculator portion of the AP exam.

On the calculator-active portion, a graphing calculator can be used to verify your answer or to find the value of a derivative at a single point without showing the analytical steps.

To find the numerical derivative at a point (e.g., check the slope in Example 2):

1. Access the numerical derivative function on your calculator. On a TI-84, this is typically math` -> `8:nDeriv(`. 2. The syntax is generally $nDeriv(function, variable, value).

2. To find the derivative of $g(x)=e^x\sin x$ at $x=0$, you would enter:

   nDeriv(e^(X)*sin(X), X, 0)

3. The calculator will return $1$, confirming the slope of the tangent line we found analytically.

This tool is extremely useful for checking your work on free-response questions or for quickly finding a slope when the analytical derivative is not required by the prompt. However, it cannot be used to find the general derivative function $g^{\prime }(x)$.

AP Exam Quick Hit

Common Question Types

• Combining with Product/Quotient/Chain Rules: You will be asked to find the derivative of a function that combines these new functions with others.

  • Example: Find $f^{\prime }(x)$ if $f(x)=\frac{\ln x}{x^2+1}$. (Requires Quotient Rule and the new rule for $\ln x$).

• Finding Tangent and Normal Lines: You will be asked to find the equation of a tangent or normal line to one of these functions at a specific point.

  • Example: Find the equation of the line tangent to the graph of $y=3e^x+\cos x$ at $x=0$.

• Finding Points with a Specific Slope: You will be asked to find the $x$-values where the tangent line to a function has a certain property, such as being horizontal.

  • Example: Find all values of $x$ in the interval $(0,2\pi )$ for which the function $f(x)=x-\sin x$ has a horizontal tangent line. (Requires setting $f^{\prime }(x)=1-\cos x$ equal to 0).

Common Mistakes

• Sign Error on Cosine's Derivative: A very frequent error is forgetting the negative sign when differentiating the cosine function. Remember, $\frac{d}{dx}(\cos x)=-\sin x$.

• Misapplying the Power Rule: Students incorrectly apply the Power Rule to $e^x$ or $\ln x$. The derivative of $e^x$ is not $x{e}^{x-1}$, and the derivative of $\ln x$ is not $1$. These functions have their own distinct rules.

• Forgetting the Product or Quotient Rule: When faced with a function like $f(x)=x\ln x$, students might incorrectly differentiate each factor separately and write $f^{\prime }(x)=1\cdot \frac{1}{x}$. You must use the Product Rule.

• Confusing Derivatives and Integrals: As you learn more calculus, it can be easy to mix up derivative and integral rules. For example, the integral of $\cos x$ is $\sin x+C$, but the derivative of $\cos x$ is $-\sin x$. Keep the operations distinct.

• Chain Rule Oversights: Forgetting to apply the Chain Rule is one of the most common errors in all of calculus. When differentiating a function like $f(x)=e^{3x}$, the derivative is not just $e^{3x}$. The correct derivative is $e^{3x}\cdot 3$. Always check if the argument of your function is simply $x$ or something more complex.