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

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

Calculus is fundamentally about the study of change, and the derivative is our primary tool for analyzing instantaneous rates of change. Having established foundational rules for differentiating algebraic functions like polynomials (the Power Rule) and combinations of functions (the Product and Quotient Rules), we now expand our toolkit to include a critical class of non-algebraic functions known as transcendental functions. This topic introduces the specific derivative rules for four of the most important functions in mathematics: the trigonometric functions $\sin x$ and $\cos x$, the natural exponential function $e^x$, and the natural logarithmic function $\ln x$.

The core idea is to commit these four specific derivative formulas to memory. These rules are not derived from the Power Rule or other algebraic methods; they are fundamental results that form the building blocks for differentiating a vast new range of functions. Mastering the calculation of these derivatives is essential for analyzing models of periodic phenomena (like waves and oscillations), exponential growth and decay, and processes where change is proportional to the current value. These four rules, when combined with previously learned rules like the Product and Quotient Rules, unlock the ability to find the derivative of complex functions involving both algebraic and transcendental components.

Key Formulas

The following four formulas are the essential knowledge for this topic. They must be memorized, as they are the basis for all calculations within this section and are fundamental to the rest of differential calculus.

1. The Derivative of Sine

   The derivative of the sine function is the cosine function.

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

2. The Derivative of Cosine

   The derivative of the cosine function is the negative sine function. Note the critical negative sign.

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

3. The Derivative of the Natural Exponential Function

   The natural exponential function, $e^x$, is unique in that it is its own derivative. The rate of change of $e^x$ at any point is equal to its value at that point.

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

4. The Derivative of the Natural Logarithm

   The derivative of the natural logarithm function, for its domain $x>0$, is the reciprocal function.

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

Understanding the Rules in Context

The four derivative rules presented are foundational "building blocks." On their own, they are straightforward applications of memory. However, their true power in calculus is revealed when they are combined with the other differentiation rules you have already learned. The AP exam will almost always assess your understanding of these rules not in isolation, but in combination with the Constant Multiple, Sum/Difference, Product, and Quotient Rules.

Consider the function $f(x)=x^2\sin x$. To differentiate this function, you cannot simply differentiate $x^2$ and $\sin x$ separately. You must recognize that $f(x)$ is a product of two functions, $u(x)=x^2$ and $v(x)=\sin x$. Therefore, the Product Rule is required, and the new rule for the derivative of $\sin x$ is a necessary component within that larger structure.

A key conceptual point for $e^x$ is its defining characteristic: its rate of change is equal to its value. This property is why $e^x$ is fundamental to modeling phenomena like compound interest, population growth, and radioactive decay, where the rate of change is directly proportional to the current amount.

For $\ln x$, it is crucial to remember the domain of the original function. The function $f(x)=\ln x$ is only defined for positive values of $x$ (i.e., $x>0$). Its derivative, $f^{\prime }(x)=\frac{1}{x}$, is defined for all non-zero $x$. However, since the derivative must exist on the domain of the original function, we consider the derivative of $\ln x$ to be $\frac{1}{x}$ specifically on the interval $(0,\infty )$. This domain consideration is a subtle but important aspect of working with logarithmic functions.

Core Concepts & Rules

• Sine and Cosine Derivatives: The derivatives of $\sin x$ and $\cos x$ are cyclic. The derivative of $\sin x$ is $\cos x$, and the derivative of $\cos x$ is $-\sin x$. Pay close attention to the negative sign in the derivative of cosine; its omission is a frequent error.

• The Unchanging Rate of $e^x$: The function $e^x$ is its own derivative. This is a unique and fundamental property. The slope of the tangent line to the graph of $y=e^x$ at any point $(a,e^a)$ is simply $e^a$.

• Logarithmic to Rational: The derivative of the natural logarithm function, $\ln x$, is the rational function $\frac{1}{x}$. This transforms a transcendental function into a simple algebraic one.

• Building Blocks: These four rules do not operate in a vacuum. They are designed to be used in conjunction with all previously learned differentiation rules (Sum, Difference, Product, Quotient, Constant Multiple). Always analyze the overall structure of a function before deciding which rules to apply. For example, for $f(x)=\frac{e^x}{x}$, you must first apply the Quotient Rule.

Step-by-Step Example 1: Basic Application

Problem: Find the derivative of the function $f(x)=4\cos x-9e^x+7\ln x$.

Goal: Calculate $f^{\prime }(x)$ by applying the Sum/Difference and Constant Multiple rules along with the new derivative rules for $\cos x$, $e^x$, and $\ln x$.

Step 1: Apply the Sum and Difference Rule

The derivative of a sum or difference of functions is the sum or difference of their derivatives. We can differentiate the function term by term.

$$f^{\prime }(x)=\frac{d}{dx}(4\cos x)-\frac{d}{dx}(9e^x)+\frac{d}{dx}(7\ln x)$$

Step 2: Apply the Constant Multiple Rule

For each term, we can move the constant coefficient outside of the differentiation operator.

$$f^{\prime }(x)=4\cdot \frac{d}{dx}(\cos x)-9\cdot \frac{d}{dx}(e^x)+7\cdot \frac{d}{dx}(\ln x)$$

Step 3: Apply the Specific Derivative Rules

Now, substitute the known derivatives for $\cos x$, $e^x$, and $\ln x$.

• $\frac{d}{dx}(\cos x)=-\sin x$

• $\frac{d}{dx}(e^x)=e^x$

• $\frac{d}{dx}(\ln x)=\frac{1}{x}$

Substituting these into our expression gives:

$$f^{\prime }(x)=4(-\sin x)-9(e^x)+7\left(\frac{1}{x}\right)$$

Step 4: Simplify the Final Expression

Combine the terms to get the final answer.

$$f^{\prime }(x)=-4\sin x-9e^x+\frac{7}{x}$$

This example demonstrates the direct application of the four new rules in a straightforward combination with other basic differentiation properties.

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

Problem: Let $g(x)=\frac{e^x}{\sin x}$. Find the slope of the normal line to the graph of $g(x)$ at $x=\frac{\pi }{2}$.

Goal: This problem requires multiple steps. First, find the derivative $g^{\prime }(x)$ using the Quotient Rule. Second, evaluate the derivative at the specific point $x=\frac{\pi }{2}$ to find the slope of the tangent line. Third, find the slope of the normal line, which is the negative reciprocal of the tangent slope.

Step 1: Identify the Need for the Quotient Rule

The function $g(x)$ is a quotient of two functions: a "high" function $h(x)=e^x$ and a "low" function $l(x)=\sin x$. We must use the Quotient Rule:

$$g^{\prime }(x)=\frac{l(x)h^{\prime }(x)-h(x)l^{\prime }(x)}{[l(x){]}^2}$$

Step 2: Find the Derivatives of the High and Low Functions

• $h(x)=e^x⟹h^{\prime }(x)=\frac{d}{dx}(e^x)=e^x$

• $l(x)=\sin x⟹l^{\prime }(x)=\frac{d}{dx}(\sin x)=\cos x$

Step 3: Substitute into the Quotient Rule Formula

Substitute the functions and their derivatives into the formula.

$$g^{\prime }(x)=\frac{(\sin x)(e^x)-(e^x)(\cos x)}{(\sin x{)}^2}$$

Step 4: Simplify the Derivative (Optional but Recommended)

We can factor out $e^x$ from the numerator to make the expression cleaner.

$$g^{\prime }(x)=\frac{e^x(\sin x-\cos x)}{{\sin }^{2}x}$$

Step 5: Evaluate the Derivative at $x=\frac{\pi }{2}$

Now, we find the slope of the tangent line, $m_{tan}$, by substituting $x=\frac{\pi }{2}$ into $g^{\prime }(x)$. We recall the unit circle values: $\sin (\frac{\pi }{2})=1$ and $\cos (\frac{\pi }{2})=0$.

$$m_{tan}=g^{\prime }\left(\frac{\pi }{2}\right)=\frac{e^{\pi /2}(\sin (\frac{\pi }{2})-\cos (\frac{\pi }{2}))}{{\sin }^2(\frac{\pi }{2})}$$

$$m_{tan}=\frac{e^{\pi /2}(1-0)}{(1{)}^2}=e^{\pi /2}$$

Step 6: Find the Slope of the Normal Line

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

$$m_{norm}=-\frac{1}{m_{tan}}=-\frac{1}{e^{\pi /2}}$$

This can also be written as $m_{norm}=-e^{-\pi /2}$.

Final Answer: The slope of the normal line to the graph of $g(x)$ at $x=\frac{\pi }{2}$ is $-e^{-\pi /2}$.

Using Your Calculator

The derivative rules for $\sin x$, $\cos x$, $e^x$, and $\ln x$ are purely analytical. The AP exam will require you to know these rules by heart and show the symbolic steps for finding the derivative, as in the examples above. A calculator cannot produce the symbolic derivative (e.g., it cannot turn $\frac{d}{dx}(x\sin x)$ into $\sin x+x\cos x$).

However, a graphing calculator is an excellent tool for verifying your work on problems that involve finding the derivative at a specific point.

Action: Use the numerical derivative function (often labeled nDeriv on TI-84 style calculators, or found under a $math$ menu) to check a calculation.

Example: Let's verify the slope of the tangent line from Example 2. We found that for $g(x)=\frac{e^x}{\sin x}$, the slope at $x=\frac{\pi }{2}$ was $e^{\pi /2}$.

Calculator Steps (TI-84 Style):

1. Press the MATH button and scroll down to 8: nDeriv( or press the shortcut ALPHAWINDOW and select 3: d/dx(.

2. The syntax will be $d/dx(expression,variable,value)$.

3. Enter the function, the variable, and the point at which you want to evaluate the derivative. Make sure your calculator is in Radian Mode for trigonometric functions.

   • $d/dx(e^{(}X)/sin(X),X,\pi /2)$

4. Press ENTER. The calculator will return a numerical approximation.

   • Result: $\mathrm{4.810477...}$

5. Now, calculate the value of your analytical answer, $e^{\pi /2}$, separately.

   • $e^{(}\pi /2)$

   • Result: $\mathrm{4.810477...}$

Since the numerical derivative from the calculator matches the value of your calculated answer, you can be confident that your derivative, $g^{\prime }(x)$, and your evaluation at $x=\frac{\pi }{2}$ are correct.

AP Exam Quick Hit

Common Question Types

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

  • Example: "If $f(x)=x^3\ln x$, find $f^{\prime }(x)$." (Requires Product Rule)

  • Example: "Find the derivative of $y=\frac{3x+2}{\cos x}$." (Requires Quotient Rule)

• Finding Tangent and Normal Lines: A classic application of the derivative is finding the equation of a line tangent or normal to a curve at a given point.

  • Example: "Find the equation of the line tangent to the graph of $f(x)=e^x+\sin x$ at $x=0$."

• Finding Horizontal Tangents: This involves setting the derivative equal to zero and solving for $x$.

  • Example: "Find all x-values on the interval $[0,2\pi ]$ where the function $f(x)=e^x\cos x$ has a horizontal tangent line."

Common Mistakes

• Forgetting the Negative on Cosine's Derivative: The most common error is stating that $\frac{d}{dx}(\cos x)=\sin x$. It is negative sine: $\frac{d}{dx}(\cos x)=-\sin x$.

• Incorrectly Applying the Power Rule: Students sometimes try to apply the power rule to $e^x$ or $\ln x$. For example, writing $\frac{d}{dx}(e^x)=x{e}^{x-1}$ is incorrect. These functions have their own specific rules that must be memorized.

• Product/Quotient Rule Neglect: Seeing a function like $y=e^x\sin x$ and finding the derivative as $y^{\prime }=e^x\cos x$. This is incorrect because it ignores the Product Rule. You must treat it as a product of two functions.

• Chain Rule Confusion (Future Topic Warning): These rules apply to $\sin x$, $\cos x$, $e^x$, and $\ln x$. They do not directly apply to composite functions like $\sin (5x)$ or $e^{x^2}$. Differentiating these requires the Chain Rule, a subsequent topic. A common early mistake is to write $\frac{d}{dx}(\cos (2x))=-\sin (2x)$, which is incomplete. Be sure you are only applying these rules to the base functions as written.