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

Welcome back to AP Calculus with Fiveable! Now that you’ve mastered finding derivatives of trigonometric functions $\sin x$ and $\cos x$, let’s cover the rest! Remembering these rules is key to simplifying your calculus journey. 🌟

💫 Derivatives of Advanced Trigonometric Functions

First, let’s have a glance at a summary table for quick reference.

It's important to note that these are only valid for angles in radians, not degrees.

Derivative of $\tan x$

The derivative of $\tan x$ is $\sec^2 x$. Let’s consider an example:

To find the derivative of this equation, differentiate $3\tan x$ and $2x^2$ individually.

Since the derivative of $\tan x$ is $\sec^2 x$, the derivative of the first part is $3\sec^2 x$. The derivative of $2x^2$ is $4x$. Hence, $f'(x) = 3\sec^2 x + 4x$.

Derivative of $\cot x$

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

We again have to differentiate the two terms separately! The derivative of $\cot x$ is $-\csc^2 x$, so the derivative of the first term is $-5\csc^2 x$. The derivative of $x$ is $1$. Therefore, $f'(x) = -5\csc^2 x + 1$ or $f'(x)=1-5\csc^2 x$.

Derivative of $\sec x$

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

Knowing the above trig derivative rule, the derivative of the first term is $2\sec x \tan x$. The derivative of $3x^3$ is $9x^2$. Thus, $f'(x) = 2\sec x \tan x + 9x^2$.

Derivative of $\csc x$

Last but not least, the derivative of $\csc x$ is $-\csc x \cot x$. For instance:

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

🏋️‍♂️ Practice Problems

Here are a couple of questions for you to get the concepts down!

❓ Advanced Trig Derivative Practice Questions

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. $h(x) = 5\cot(x)$

💡 Before we reveal the answers, remember to use the chain rule, sum rule, and quotient rules.

🤔 Advanced Trig Derivative Practice Solutions

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

🌟 Closing

Practice these rules, and you’ll soon find them as intuitive as the basic derivatives! Keep up the great work. 🌈