2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

The four derivatives you need are $\frac{d}{dx}\tan x=\sec^2 x$, $\frac{d}{dx}\cot x=-\csc^2 x$, $\frac{d}{dx}\sec x=\sec x\tan x$, and $\frac{d}{dx}\csc x=-\csc x\cot x$. You can either memorize them or derive each one by rewriting the function as a ratio of $\sin x$ and $\cos x$ and applying the quotient rule. For AP Calculus, keep the negative signs on cotangent and cosecant derivatives.

Why This Matters for the AP Calculus Exam

Tangent, cotangent, secant, and cosecant show up all over differentiation problems, so knowing their derivatives quickly saves time on both multiple-choice and free-response work. This topic also reinforces the bigger idea that you can rewrite a function using identities and then apply rules you already know, like the quotient rule and product rule. Once you combine these with the chain rule, you can differentiate composite expressions like $\tan(6x)$ or $\sec(ax+b)$, which appear in later units on motion, related rates, and curve analysis.

Key Takeaways

• Learn the four derivatives: $\sec^2 x$ for $\tan x$, $-\csc^2 x$ for $\cot x$, $\sec x\tan x$ for $\sec x$, and $-\csc x\cot x$ for $\csc x$.
• The two "co" functions (cotangent and cosecant) have negative derivatives. Pair that fact with the matching identities to keep signs straight.
• You can derive every rule by rewriting in terms of $\sin x$ and $\cos x$ and using the quotient rule, so memorizing is optional.
• These derivatives only work when the angle is in radians.
• Combine these with the sum, product, quotient, and chain rules to handle longer expressions.
• Rewriting a trig expression with an identity first often makes the derivative much easier.

Derivatives of Tangent, Cotangent, Secant, and Cosecant

Here is a quick-reference table for the four derivatives.

These hold only for angles measured in radians, not degrees.

The reason these rules work is that each function can be written using $\sin x$ and $\cos x$, so you can differentiate with the quotient rule. For example, since $\tan x = \frac{\sin x}{\cos x}$, the quotient rule gives:

$\frac{d}{dx}\tan x = \frac{\cos x \cdot \cos x - \sin x \cdot (-\sin x)}{\cos^2 x} = \frac{\cos^2 x + \sin^2 x}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x$

You can build the other three the same way using $\cot x = \frac{\cos x}{\sin x}$, $\sec x = \frac{1}{\cos x}$, and $\csc x = \frac{1}{\sin x}$.

Derivative of $\tan x$

The derivative of $\tan x$ is $\sec^2 x$. Consider:

$f(x)=3\tan x+2x^2$

Differentiate $3\tan x$ and $2x^2$ separately. The derivative of $\tan x$ is $\sec^2 x$, so the first part becomes $3\sec^2 x$. The derivative of $2x^2$ is $4x$. So $f'(x) = 3\sec^2 x + 4x$.

Derivative of $\cot x$

The derivative of $\cot x$ is $-\csc^2 x$. For example:

$f(x)=5\cot x+x$

Differentiate term by term. The derivative of $\cot x$ is $-\csc^2 x$, so the first term becomes $-5\csc^2 x$. The derivative of $x$ is $1$. So $f'(x) = 1 - 5\csc^2 x$.

Derivative of $\sec x$

The derivative of $\sec x$ is $\sec x \tan x$. For example:

$f(x)=2\sec x+3x^3$

The first term becomes $2\sec x \tan x$, and the derivative of $3x^3$ is $9x^2$. So $f'(x) = 2\sec x \tan x + 9x^2$.

Derivative of $\csc x$

The derivative of $\csc x$ is $-\csc x \cot x$. For example:

$f(x)=4\csc x+7x^2$

The first part becomes $-4\csc x \cot x$, and the derivative of $7x^2$ is $14x$. So $f'(x) = -4\csc x \cot x + 14x$.

How to Use This on the AP Calculus Exam

Problem Solving

Identify which trig function you have, then apply the matching derivative. When the function is bundled with polynomial or other terms, differentiate term by term using the sum and constant multiple rules.

Common Trap

When the inside of the trig function is more than just $x$, you need the chain rule. For example, $\frac{d}{dx}\tan(6x) = 6\sec^2(6x)$, and $\frac{d}{dx}\sec(ax+b) = a\sec(ax+b)\tan(ax+b)$. Forgetting to multiply by the derivative of the inside is one of the most common mistakes.

Rewrite First

If an expression looks messy, try rewriting with an identity before differentiating. A quotient of trig functions can sometimes simplify into a single function, which makes the derivative shorter and lowers your chance of an algebra error. Clear structure is important for clear exam work, especially on free-response.

Practice Problems

Find the derivatives for the following problems.

1. $f(x) = 2 \tan(x) + \sec(x)$
2. $f(x) = \frac{\cot(x)}{\csc(x)}$
3. $g(x) =\tan^2(6x)$
4. $h(x) = 5\cot(x)$

Use the chain rule, sum rule, and quotient rule where needed.

Solutions

1. $f'(x) = 2 \sec^2(x) + \sec(x) \tan(x)$
2. $f'(x)=-\csc^2(x)$ (rewrite $\frac{\cot x}{\csc x}$ as $\cos x$ first, since $\frac{\cot x}{\csc x} = \frac{\cos x / \sin x}{1/\sin x} = \cos x$, then differentiate)
3. $g'(x)=2\tan(6x)\cdot 6\sec^2(6x) = 12\tan(6x)\sec^2(6x)$
4. $h'(x)=-5\csc^2(x)$

These problems combine all the derivative rules from this unit. If you need a refresher, check these out:

2.5 Applying the Power Rule

2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple

2.7 Derivatives of cos x, sinx, e^x, and ln x

2.8 The Product Rule

2.9 The Quotient Rule

Common Misconceptions

• Misconception: The derivatives of cot x and csc x are positive, while tan x and sec x are negative.
  Actually: The derivatives of $\cot x$ and $\csc x$ are negative, while $\tan x$ and $\sec x$ are positive. Mixing up the signs is a frequent error.
• Misconception: The derivative of $\sec x$ is $\sec^2 x$.
  Actually: $\frac{d}{dx}\sec x$ is $\sec x\tan x$, not $\sec^2 x$. Keep the secant and tangent rules separate.
• Misconception: These derivative rules hold for degrees.
  Actually: These derivative rules assume the angle is in radians. They do not hold for degrees.
• Misconception: The chain rule is not required whenever the angle is a function of $x$.
  Actually: The chain rule is required whenever the angle is a function of $x$, not just $x$ by itself. For $\tan(6x)$, you must multiply by $6$.
• Misconception: Rewriting with an identity changes the function, so it is not a safe step.
  Actually: Rewriting with an identity does not change the function, so it is a safe step. Use it to simplify before differentiating instead of forcing the quotient rule through a long expression.

Related AP Calculus Guides

• Unit 2 Overview: Differentiation
• 2.1 Defining Average and Instantaneous Rates of Change at a Point
• 2.2 Defining the Derivative of a Function and Using Derivative Notation
• 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
• 2.3 Estimating Derivatives of a Function at a Point
• 2.7 Derivatives of cos x, sinx, e^x, and ln x