How to Find Derivatives

Download Article

The formulas you need to work it out on a calculator or by hand

Reviewed by Joseph Meyer

Last Updated: September 10, 2023 Fact Checked

Download Article

Preliminaries

|

Basic Techniques

|

Using a Calculator

|

Video

|

Q&A

|

Tips

|

Warnings

The derivative is an operator that finds the instantaneous rate of change of a quantity, usually a slope. Derivatives can be used to obtain useful characteristics about a function, such as its extrema and roots. Finding the derivative from its definition can be tedious, but there are many techniques to bypass that and find derivatives more easily.

Method 1

Method 1 of 3:

Preliminaries

Download Article

{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/2\/24\/Take-Derivatives-Step-1.jpg\/v4-460px-Take-Derivatives-Step-1.jpg","bigUrl":"\/images\/thumb\/2\/24\/Take-Derivatives-Step-1.jpg\/v4-728px-Take-Derivatives-Step-1.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\">\u00a9 2024 wikiHow, Inc. All rights reserved. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. This image may not be used by other entities without the express written consent of wikiHow, Inc. \n<\/p> \n<\/p><\/div>"}

1
Understand the definition of the derivative. While this will almost never be used to actually take derivatives, an understanding of this concept is vital nonetheless. ^{[1]
X
Research source}
- Recall that the linear function is of the form $y=mx+b.$ To find the slope $m$ of this function, two points on the line are taken, and their coordinates are plugged into the relation $m={\frac {y_{{2}}-y_{{1}}}{x_{{2}}-x_{{1}}}}.$ Of course, this can only be used with linear graphs.
- For nonlinear functions, the line will be curved, so taking the difference of two points can only give the average rate of change between them. The line that intersects these two points is called the secant line, with a slope $m={\frac {f(x+\Delta x)-f(x)}{\Delta x}},$ where $\Delta x=x_{{2}}-x_{{1}}$ is the change in $x,$ and we have replaced $y$ with $f(x).$ This is the same equation as the one before.
- The concept of the derivatives comes in when we take the limit $\Delta x\to 0.$ When this happens, the distance between the two points shrinks, and the secant line better approximates the rate of change of the function. When we do send the limit to 0, we end up with the instantaneous rate of change and obtain the slope of the tangent line to the curve (see animation above). ^{[2]
 X
 Research source} Then, we end up with the definition of the derivative, where the prime symbol denotes the derivative of the function $f.$
 - $f^{{\prime }}(x)=\lim _{{\Delta x\to 0}}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}$
- Finding the derivative from this definition stems from expanding the numerator, canceling, and then evaluating the limit, since immediately evaluating the limit will give a 0 in the denominator.
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/e\/ea\/Take-Derivatives-Step-2.jpg\/v4-460px-Take-Derivatives-Step-2.jpg","bigUrl":"\/images\/thumb\/e\/ea\/Take-Derivatives-Step-2.jpg\/v4-728px-Take-Derivatives-Step-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\">\u00a9 2024 wikiHow, Inc. All rights reserved. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. This image may not be used by other entities without the express written consent of wikiHow, Inc. \n<\/p> \n<\/p><\/div>"}

2
Understand the derivative notation. There are two common notations for the derivative, though there are others. ^{[3]
X
Research source}
- Lagrange's Notation. In the previous step, we used this notation to denote the derivative of a function $f(x)$ by adding a prime symbol.
 - $f^{{\prime }}(x)$
 - This notation is pronounced " $f$ prime of $x.$ " To form higher order derivatives, simply add another prime symbol. When derivatives of fourth or higher order are taken, the notation becomes $f^{{(4)}}(x),$ where this represents the fourth derivative.
- Leibniz's Notation. This is the other commonly used notation, and we will use it in the rest of the article.
 - ${\frac {{\mathrm {d}}f}{{\mathrm {d}}x}}$
 - (For shorter expressions, the function can be placed in the numerator.) This notation literally means "the derivative of $f$ with respect to $x.$ " It may be helpful to think of it as ${\frac {\Delta y}{\Delta x}}$ for values of $x$ and $y$ that are infinitesimally different from each other. When using this notation for higher derivatives, you must write ${\frac {{\mathrm {d}}^{{2}}f}{{\mathrm {d}}x^{{2}}}},$ where this represents the second derivative.
 - (Note that there "should" be parentheses in the denominator, but no one ever writes them, since everyone understands what we mean without them anyway.)

Method 2

Method 2 of 3:

The Chain Rule

{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/d\/da\/Take-Derivatives-Step-11.jpg\/v4-460px-Take-Derivatives-Step-11.jpg","bigUrl":"\/images\/thumb\/d\/da\/Take-Derivatives-Step-11.jpg\/v4-728px-Take-Derivatives-Step-11.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\">\u00a9 2024 wikiHow, Inc. All rights reserved. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. This image may not be used by other entities without the express written consent of wikiHow, Inc. \n<\/p> \n<\/p><\/div>"}

1
Use the chain rule ^{[7]

X
Research source} for nested functions. For example, consider the scenario where $z(y)$ is a differentiable function of $y$ and $y(x)$ is a differentiable function of $x.$ Then there is a composite function $z(y(x)),$ or $z$ as a function of $x,$ that we can take the derivative of.
- ${\frac {{\mathrm {d}}}{{\mathrm {d}}x}}z(y(x))={\frac {{\mathrm {d}}z}{{\mathrm {d}}y}}{\frac {{\mathrm {d}}y}{{\mathrm {d}}x}}$
- As with the product rule, this works with any number of functions; hence the "chain" rule. Here, an easy way to see how this works is if one imagines a ${\frac {{\mathrm {d}}y}{{\mathrm {d}}y}}$ inserted between ${\frac {{\mathrm {d}}z}{{\mathrm {d}}x}}.$
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/2\/26\/Take-Derivatives-Step-12.jpg\/v4-460px-Take-Derivatives-Step-12.jpg","bigUrl":"\/images\/thumb\/2\/26\/Take-Derivatives-Step-12.jpg\/v4-728px-Take-Derivatives-Step-12.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\">\u00a9 2024 wikiHow, Inc. All rights reserved. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. This image may not be used by other entities without the express written consent of wikiHow, Inc. \n<\/p> \n<\/p><\/div>"}

2
Consider the function $f(x)=(2x^{{4}}-x)^{3}$ . Notice that this function can be decomposed into two elementary functions, $g(x)=2x^{{4}}-x$ and $h(g)=g^{{3}}.$ Then, we want to find the derivative of the composition $f(x)=h(g(x)).$
- Use the chain rule ${\frac {{\mathrm {d}}}{{\mathrm {d}}x}}h(g(x))={\frac {{\mathrm {d}}h}{{\mathrm {d}}g}}{\frac {{\mathrm {d}}g}{{\mathrm {d}}x}}.$ We have now written the derivative in terms of derivatives that are easier to take. Then,
- ${\frac {{\mathrm {d}}}{{\mathrm {d}}x}}h(g(x))=3(2x^{4}-x)^{2}(8x^{3}-1).$
- With practice, you will see that applying the chain rule is easiest if you "peel away the onion." The first layer is everything inside the parentheses, cubed. The second layer is the function inside the parentheses. When dealing with more complex functions, this way of thinking helps to keep yourself on track and not get lost in what functions are taken with respect to what variables, etc.

Other Important Derivatives

{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/8\/88\/Take-Derivatives-Step-13.jpg\/v4-460px-Take-Derivatives-Step-13.jpg","bigUrl":"\/images\/thumb\/8\/88\/Take-Derivatives-Step-13.jpg\/v4-728px-Take-Derivatives-Step-13.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\">\u00a9 2024 wikiHow, Inc. All rights reserved. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. This image may not be used by other entities without the express written consent of wikiHow, Inc. \n<\/p> \n<\/p><\/div>"}

See this article for a full treatment on implicit differentiation. Understanding the chain rule is a must in order to implicitly differentiate.
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/7\/7d\/Take-Derivatives-Step-14.jpg\/v4-460px-Take-Derivatives-Step-14.jpg","bigUrl":"\/images\/thumb\/7\/7d\/Take-Derivatives-Step-14.jpg\/v4-728px-Take-Derivatives-Step-14.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\">\u00a9 2024 wikiHow, Inc. All rights reserved. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. This image may not be used by other entities without the express written consent of wikiHow, Inc. \n<\/p> \n<\/p><\/div>"}

2
See this article for a full treatment on differentiating exponential functions.
- ${\frac {{\mathrm {d}}}{{\mathrm {d}}x}}e^{{x}}=e^{{x}}$
- ${\frac {{\mathrm {d}}}{{\mathrm {d}}x}}a^{{x}}=a^{{x}}\ln a$
- ${\frac {{\mathrm {d}}}{{\mathrm {d}}x}}\ln x={\frac {1}{x}}$
- ${\frac {{\mathrm {d}}}{{\mathrm {d}}x}}\log _{{a}}x={\frac {1}{x\ln a}}$
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/4\/47\/Take-Derivatives-Step-15.jpg\/v4-460px-Take-Derivatives-Step-15.jpg","bigUrl":"\/images\/thumb\/4\/47\/Take-Derivatives-Step-15.jpg\/v4-728px-Take-Derivatives-Step-15.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\">\u00a9 2024 wikiHow, Inc. All rights reserved. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. This image may not be used by other entities without the express written consent of wikiHow, Inc. \n<\/p> \n<\/p><\/div>"}

3
Memorize basic trigonometric derivatives and how to derive them. ^{[8]
X
Research source}
- ${\frac {{\mathrm {d}}}{{\mathrm {d}}x}}\sin x=\cos x$
- ${\frac {{\mathrm {d}}}{{\mathrm {d}}x}}\cos x=-\sin x$
- ${\frac {{\mathrm {d}}}{{\mathrm {d}}x}}\tan x=\sec ^{{2}}x$
- ${\frac {{\mathrm {d}}}{{\mathrm {d}}x}}\cot x=-\csc ^{{2}}x$
- ${\frac {{\mathrm {d}}}{{\mathrm {d}}x}}\sec x=\sec x\tan x$
- ${\frac {{\mathrm {d}}}{{\mathrm {d}}x}}\csc x=-\csc x\cot x$

Method 3

Method 3 of 3:

Using a Calculator

Download Article

1
Press Alpha F2 . This will open the “Window” key, where you’ll see lots of options. Scroll over to the FUNC tab if you aren’t there already. ^{[9]
X
Research source}
- These instructions are for new models of the TI-84 and the TI-84 Plus. Older models may be slightly different.
2

Select nDeriv( . It’s the third option on the list. When you get to it, you can press “enter” to select it. ^{[10]
X
Research source}
3
Enter your formula into the equation. When you hit the derivative option, your calculator will give you a blank equation that looks like this: $(d/d[])([])|x=[]$ . Go ahead and enter your specific numbers into the equation. ^{[11]
X
Research source}
- For example, if you were finding the derivative of the function $x^{2}$ where $x=2$ , you’d enter $(d/dx)(x^{2})|x=2$ .
- If you have an equation plotted in the Y plots of your calculator, you can enter those into a blank field by pressing vars > Y-VARS > Function .
4
Hit “enter” to find the derivative. Once you have all of your numbers entered, you can select “enter” on your calculator to get your answer. It will (hopefully) give you your answer in an easy to understand whole number. ^{[12]
X
Research source}
- For example, in the equation above, the derivative is 4.

Community Q&A

Search

Add New Question

Question

What is the derivative of x^0.09?

Pimemorized

Top Answerer

By the power rule, you would first multiply the whole equation by the exponent, which is 0.09 so you get 0.09x^0.09. Then, you subtract the power by one, so it becomes 0.09x^(-.91).

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 1 Helpful 10
Question

What is the second derivative of 1/x?

Community Answer

2 x^-3

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 17 Helpful 29
Question

What is the derivative of a logarithmic function?

Pimemorized

Top Answerer

The derivative of a log function is the derivative of the function divided by the function itself. For example, the derivative of log(x) would be the derivative of x is 1 divided by x, and so log(x) = 1/x.

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 5 Helpful 6

See more answers

Ask a Question

200 characters left

Include your email address to get a message when this question is answered.

Submit

Video

Tips

Every technique outlined in this article on calculating derivatives can be verified by a proper use of the definition of the derivative. If, for example, the power rule seems sketchy to you, try and recover the formula using the definition.

Thanks
Helpful 0 Not Helpful 0
Practice the product rule, chain rule, and especially implicit differentiation, as these are more difficult to differentiate and are widely used outside mathematics.

Thanks
Helpful 0 Not Helpful 0

Submit a Tip

All tip submissions are carefully reviewed before being published

Name

Please provide your name and last initial

Submit

Thanks for submitting a tip for review!

Warnings

Some students will be tempted to use programs on their calculators to take derivatives. While these programs are very useful for confirming your answers, you should not rely on these. Make sure you understand the concepts of deriving and are able to do it yourself.

Thanks
Helpful 2 Not Helpful 0

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

Log in

How to Find Derivatives

Steps

Preliminaries

Basic Techniques

Using the Definition

The Power Rule

Higher Order Derivatives

The Product and Quotient Rules

The Chain Rule

Other Important Derivatives

Using a Calculator

Community Q&A

Video

Tips

Warnings

You Might Also Like

References

About This Article

Reader Success Stories

Did this article help you?

Quizzes

You Might Also Like

Featured Articles

Trending Articles

Featured Articles

Featured Articles

Watch Articles

Trending Articles

Quizzes

Explore Categories