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

The Remaining Trig Derivatives You Must Know

All of these assume x is in radians.

$$\frac{d}{dx}(\tan x) = \sec^2 x$$

$$\frac{d}{dx}(\cot x) = -\csc^2 x$$

$$\frac{d}{dx}(\sec x) = \sec x \tan x$$

$$\frac{d}{dx}(\csc x) = -\csc x \cot x$$

A quick memory pattern:

• tan → sec²
• cot → −csc²
• sec → sec·tan
• csc → −csc·cot

Only cot and csc have negatives.

If the input is not just $x$, use the chain rule:

$$\frac{d}{dx}[\tan(g(x))] = \sec^2(g(x)) \cdot g'(x)$$

Example:

$$\frac{d}{dx}[\sec(4x^3)] = \sec(4x^3)\tan(4x^3)\cdot 12x^2$$

That inside derivative is where people drop points.

Rewriting with Identities Before Differentiating

You do not always have to use the memorized derivative formula. Sometimes rewriting makes the problem cleaner.

Key identities:

• $\tan x = \frac{\sin x}{\cos x}$
• $\cot x = \frac{\cos x}{\sin x}$
• $\sec x = \frac{1}{\cos x}$
• $\csc x = \frac{1}{\sin x}$

Example:

$$f(x) = \frac{\cot x}{\csc x}$$

Rewrite:

$$\frac{\cos x / \sin x}{1/\sin x}$$

Multiply by the reciprocal:

$$= \cos x$$

Now the derivative is just:

$$f'(x) = -\sin x$$

Way easier than quotient rule.

On tests, especially no-calculator sections, simplifying first can save time and reduce algebra mistakes.

Using Product Rule with These Functions

Trig functions rarely appear alone. You’ll often see something like:

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

This is a product, so:

$$f'(x) = 2x\tan x + x^2\sec^2 x$$

Notice how we plugged in the derivative of tan immediately.

Another example:

$$f(x) = \csc x \ln x$$

$$f'(x) = (-\csc x \cot x)\ln x + \csc x \cdot \frac{1}{x}$$

Keep the trig derivative clean before worrying about simplification.

Using Quotient Rule

If rewriting doesn’t simplify nicely, use quotient rule:

$$\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{f'g - fg'}{g^2}$$

Example:

$$f(x) = \frac{\tan x}{x^3}$$

$$f'(x) = \frac{\sec^2 x \cdot x^3 - \tan x \cdot 3x^2}{x^6}$$

You could factor later, but structurally this is correct and earns full credit.

On free-response, structure matters more than perfect simplification.

Chain Rule Situations

This is where most errors happen.

If you see:

• $\tan(5x)$
• $\sec(x^2+1)$
• $\csc(3x-4)$
• $\tan^3(2x)$

There is an inner function.

Example:

$$\frac{d}{dx}[\tan(5x)] = \sec^2(5x)\cdot 5$$

Power + trig + chain together:

$$\frac{d}{dx}[\tan^3(2x)]$$

Think of it as $(\tan(2x))^3$:

$$= 3\tan^2(2x)\cdot \sec^2(2x)\cdot 2$$

Three layers:

1. Power rule
2. Trig derivative
3. Inner derivative

BC questions often stack rules like this on purpose.

When to Simplify First

Before you differentiate, pause and scan.

Simplify first if:

• Trig ratios cancel.
• Expressions reduce to sin or cos.
• Algebra shortens the problem.

Differentiate first if:

• It’s already clearly a product or quotient.
• Rewriting makes it longer.

That small decision can save you a lot of time under pressure.

Common Mistakes

• Writing $\frac{d}{dx}(\tan x) = \sec x$. It is sec² x.
• Forgetting the negative on cot or csc.
• Missing the inner derivative in $\tan(7x)$.
• Using degrees instead of radians.
• Dropping parentheses in expressions like $\sec^2(3x)$.