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You CAN Ace Calculus

17calculus > derivatives > trig derivatives

prerequisites for basic trig derivatives

basic derivative rules

power rule

product rule

quotient rule

additional prerequisites for derivatives involving chain rule

chain rule

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Trigonometric and Inverse Trigonometric Derivatives

1. basic trigonometric derivatives
2. basic inverse trigonometric derivatives
3. trigonometric derivatives using the chain rule
4. inverse trigonometric derivatives using the chain rule
If you haven't studied the chain rule yet, you can learn about trigonometric derivatives basics in the first two sections and then come back once you have learned the chain rule.
Also, if you are studying only trig derivatives but not inverse trig derivatives yet, you can skip the inverse trig sections without missing anything.

Basic Trig Derivatives (no chain rule required)

This first section discuss the basics of trig derivatives and does not require you to know the chain rule. Here are the basic rules. If you work enough practice problems, you won't need to memorize them since you will just know them.

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

The first two, sine and cosine, are pretty straightforward since they are ALMOST inverses of each other. You just need to remember the negative sign on the second one.The rest of them can be derived from the sine and cosine rules using the product rule, quotient rule and basic trigonometric identities. Here is a good video showing this derivation.

 PatrickJMT: Deriving the Derivative Formulas for Tangent, Cotangent, Secant, Cosecant

Notice how the derivatives seem to be in similar pairs. Take a few minutes and look for patterns and similarities as well as differences in all three sets. Take special notice of when a minus sign appears.

Okay, let's take a minute and watch a quick video clip for another perspective on how to remember these derivatives.

 Krista King Math: Derivatives of TRIG FUNCTIONS

Here are some practice problems. Calculate the derivative of these functions.

 Basic Problems

Practice 1

$$\displaystyle{x^4\tan(x)}$$

solution

Practice 2

$$\displaystyle{f(x)=\cos(x)\sin(x)}$$

solution

Practice 3

$$\displaystyle{\frac{\sin(x)}{1+\cos(x)}}$$

solution

Practice 4

$$\displaystyle{y=\frac{\tan(x)}{x^{3/2}+5x}}$$

solution

 Intermediate Problems

Practice 5

$$\displaystyle{\frac{3}{x}\cot(x)}$$

solution

Basic Inverse Trig Derivatives (no chain rule required)

You might expect the derivatives for inverse trig functions to be similar to derivatives for trig functions but they are not. They are very different. But, again, they appear in similar pairs.

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Important Things To Notice
1. Strangely enough, the derivative of inverse trig functions do not contain any trig or inverse trig terms. We include some videos below showing the derivation of these equations.
2. Each pair of inverse trig derivatives are very closely related, even closer than with trig derivatives. Each pair is the same EXCEPT for a negative sign. So, for example, $$[\arccos(t)]' = -[\arcsin(t)]'$$.
3. In the case of the third pair, $$[\arcsec(t)]'$$ and $$[\arccsc(t)]'$$, the denominators contain an absolute value term, $$\abs{t}$$, which is important. Do not leave off the absolute value signs unless you explicitly state that $$t$$ is always positive. Keeping them is always the safe way to make sure you are correct. ( Of course, you need to check with your instructor to see what they require. )
4. Remember that the notation $$\sin^{-1}(t)$$ actually means $$\arcsin(t)$$, not $$1/\sin(t)$$.

Before we go on, let's watch some videos showing how the derivatives above come about. It seems strange that when we take the derivative of a function involving inverse trig functions, there are no longer any trig or inverse trig terms. These next videos show how to get the derivatives. Watching these will help you understand trig and inverse trig in more depth.

 PatrickJMT: The Derivative of Inverse Sine or y = arcsin(x)
 MIP4U: Proof - The Derivative of f(x)=arccos(x): d/dx[arccos(x)]
 PatrickJMT: Deriving the Derivative of Inverse Tangent or y = arctan (x)
 MIP4U: Proof - The Derivative of f(x)=arccot(x): d/dx[arccot(x)]
 MIP4U: Proof - The Derivative of f(x)=arcsec(x): d/dx[arcsec(x)]
 MIP4U: Proof - The Derivative of f(x)=arccsc(x): d/dx[arccsc(x)]

Trig Derivatives Using The Chain Rule

This section requires understanding of the chain rule and the previous sections on the basics of trig derivatives. Except for the most basic problems, you will find most problems require the use of the chain rule.
As we discuss on the chain rule page, we think the easiest way to work problems that require the chain rule is to start on the outside and work your way in. Doing it the other direction, requires you to know where to start and what is actually on the inside. Especially when starting out, this is more complicated in our opinion. ( Of course, you need to consult your instructor to find out what they require. )

The best way to work these derivatives is to use substitution, especially while you are still learning. Later, you can do it in your head. In the examples on this page, we will write out the substitution to help you see it.

Let's start out by watching a video. This is a great video explaining the idea of the chain rule when a trig function is involved.

 Krista King Math: Chain rule for derivatives, with trig functions

Okay, let's work some examples. If you feel up to it, try to work these before opening the solutions.

 Example 1 - - Evaluate $$[\sin(x^2)]'$$
 Example 2 - - Evaluate $$[\sec(x^2)]'$$

This next example uses the product rule and derivative of exponentials. If you have not studied both of these topics, you can skip this example and come back to it once you have. We will also show you how to work a nested chain rule problem.

 Example 3 - - Evaluate $$[\csc(2xe^{15x})]'$$

Here are some practice problems. Unless otherwise instructed, calculate the derivative of these functions.

 Basic Problems

Practice 6

$$\displaystyle{ 7 \sin(x^2+1) }$$

solution

Practice 7

$$\displaystyle{ y=\sin^3(x) \tan(4x) }$$

solution

Practice 8

$$\displaystyle{ y = (2x^5-3) \sin(7x) }$$

solution

Practice 9

$$\tan(x^5 + 2x^3 - 12x)$$

solution

 Intermediate Problems

Practice 10

$$\displaystyle{ y = (1+\cos^2(7x))^3 }$$

solution

Practice 11

$$y=\cot^4(x/2)$$

solution

Practice 12

Calculate the third derivative of $$f(x)=\tan(3x)$$.

solution

Practice 13

$$\displaystyle{ y=\left[ \tan \left( \sin \left( \sqrt{x^2+8x}\right) \right) \right]^5 }$$

solution

Practice 14

$$\displaystyle{ y=x \sin(1/x) + \sqrt[4]{(1-3x)^2 + x^5} }$$

solution

Inverse Trig Derivatives Using The Chain Rule

As mentioned in the previous section, except for the simplest problems, you will need to use the chain rule for most inverse trig derivatives. Let's look at an example that parallels some of the examples in the previous section.

 Example 4 - - Evaluate $$[\arcsin(x^2)]'$$

3