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Type any expression to check your own work, or use one of the presets below.
21 problems, grouped by rule
Try each one on paper first, then open the answer. Click a problem to load it into the calculator with every step shown.
Warm-up — one rule each
Tests: Power, constant multiple, sum
- \(\frac{d}{dx}\left[5 x^{4}\right]\)
- \(\frac{d}{dx}\left[x^{3} - 4 x + 7\right]\)
- \(\frac{d}{dx}\left[\frac{3}{x^{2}}\right]\)
- \(\frac{d}{dx}\left[2 \sqrt{x}\right]\)
- \(\frac{d}{dx}\left[x^{5} + x^{\frac{1}{3}}\right]\)
Product and quotient
Tests: Product, quotient
- \(\frac{d}{dx}\left[x^{2} \sin\left(x\right)\right]\)
- \(\frac{d}{dx}\left[x e^{x}\right]\)
- \(\frac{d}{dx}\left[\frac{x^{2} + 1}{x - 1}\right]\)
- \(\frac{d}{dx}\left[\sin\left(x\right) \cos\left(x\right)\right]\)
- \(\frac{d}{dx}\left[\frac{\ln\left(x\right)}{x}\right]\)
Chain rule
Tests: Chain
- \(\frac{d}{dx}\left[\sin\left(3 x\right)\right]\)
- \(\frac{d}{dx}\left[\left(2 x + 1\right)^{5}\right]\)
- \(\frac{d}{dx}\left[e^{x^{2}}\right]\)
- \(\frac{d}{dx}\left[\ln\left(1 + x^{2}\right)\right]\)
- \(\frac{d}{dx}\left[\sqrt{x^{2} + 1}\right]\)
Mixed — recognise the rule first
Tests: All of them
- \(\frac{d}{dx}\left[x^{2} e^{3 x}\right]\)
- \(\frac{d}{dx}\left[\tan\left(x^{2}\right)\right]\)
- \(\frac{d}{dx}\left[\frac{x}{x^{2} + 1}\right]\)
- \(\frac{d}{dx}\left[e^{x} \cos\left(x\right)\right]\)
- \(\frac{d}{dx}\left[\ln\left(\sin\left(x\right)\right)\right]\)
- \(\frac{d}{dx}\left[x^{x}\right]\)
33 standard derivatives worth knowing
These are the forms that appear inside almost every exam question. Each family links to the rule page that explains it, with five worked examples and a graph.
Powers and roots
| f(x) | f′(x) | Open |
|---|---|---|
| \(x^{2}\) | \(2 x\) | steps |
| \(x^{3}\) | \(3 x^{2}\) | steps |
| \(x^{n}\) | \(n x^{n - 1}\) | steps |
| \(\sqrt{x}\) | \(\frac{1}{2 \sqrt{x}}\) | steps |
| \(\sqrt[3]{x}\) | \(\frac{1}{3 x^{\frac{2}{3}}}\) | steps |
| \(\frac{1}{x}\) | \(-\frac{1}{x^{2}}\) | steps |
| \(\frac{1}{x^{2}}\) | \(-\frac{2}{x^{3}}\) | steps |
| \(x^{\frac{2}{3}}\) | \(\frac{2}{3 \sqrt[3]{x}}\) | steps |
Trigonometric
| f(x) | f′(x) | Open |
|---|---|---|
| \(\sin\left(x\right)\) | \(\cos\left(x\right)\) | steps |
| \(\cos\left(x\right)\) | \(-\sin\left(x\right)\) | steps |
| \(\tan\left(x\right)\) | \(\sec^{2}\left(x\right)\) | steps |
| \(\sec\left(x\right)\) | \(\sec\left(x\right) \tan\left(x\right)\) | steps |
| \(\csc\left(x\right)\) | \(-\cot\left(x\right) \csc\left(x\right)\) | steps |
| \(\cot\left(x\right)\) | \(-\csc^{2}\left(x\right)\) | steps |
Exponential and logarithmic
| f(x) | f′(x) | Open |
|---|---|---|
| \(e^{x}\) | \(e^{x}\) | steps |
| \(e^{2 x}\) | \(2 e^{2 x}\) | steps |
| \(2^{x}\) | \(2^{x} \ln\left(2\right)\) | steps |
| \(a^{x}\) | \(a^{x} \ln\left(a\right)\) | steps |
| \(\ln\left(x\right)\) | \(\frac{1}{x}\) | steps |
| \(\log_{2}\left(x\right)\) | \(\frac{1}{x \ln\left(2\right)}\) | steps |
| \(x \ln\left(x\right)\) | \(\ln\left(x\right) + 1\) | steps |
Hyperbolic
| f(x) | f′(x) | Open |
|---|---|---|
| \(\sinh\left(x\right)\) | \(\cosh\left(x\right)\) | steps |
| \(\cosh\left(x\right)\) | \(\sinh\left(x\right)\) | steps |
| \(\tanh\left(x\right)\) | \(\operatorname{sech}^{2}\left(x\right)\) | steps |
| \(\operatorname{arsinh}\left(x\right)\) | \(\frac{1}{\sqrt{x^{2} + 1}}\) | steps |
| \(\operatorname{artanh}\left(x\right)\) | \(\frac{1}{-x^{2} + 1}\) | steps |
Other standard forms
| f(x) | f′(x) | Open |
|---|---|---|
| \(\left|x\right|\) | \(\operatorname{sgn}\left(x\right)\) | steps |
| \(x \sin\left(x\right)\) | \(x \cos\left(x\right) + \sin\left(x\right)\) | steps |
| \(\frac{\sin\left(x\right)}{x}\) | \(\frac{x \cos\left(x\right) - \sin\left(x\right)}{x^{2}}\) | steps |
| \(x^{x}\) | \(x^{x} \left(\ln\left(x\right) + 1\right)\) | steps |
Choosing the right rule
The hard part of a derivative question is almost never the algebra — it is deciding which rule the expression calls for. Work in this order and the decision becomes mechanical:
- Is it a sum? Split it. Differentiate each term on its own and add the results.
- Is there a constant factor? Pull it out front and forget about it until the end.
- Is it a product or a quotient of two things that both contain \(x\)? Product or quotient rule.
- Is one function inside another? Chain rule — and keep the inside unchanged while you differentiate the outside.
- Otherwise it is a standard form: look it up in the table above.
Most expressions need two or three of these in sequence. \(x^{2}e^{3x}\) is a product whose second factor needs the chain rule; \(\ln(\sin x)\) is a chain whose inside is a standard form.
Habits that stop mistakes
- Write the rule's name next to each line as you use it. If you cannot name it, you are guessing.
- Check the shape of your answer before the details: a product rule answer has two terms, a quotient rule answer is one fraction.
- Differentiating does not have to simplify. \(e^{x}(1+x)\) and \(e^{x} + xe^{x}\) are the same answer.
- Sanity-check on a graph: your \(f'\) must cross zero wherever \(f\) turns. The graph calculator shows both at once.
- When you are stuck, do it from first principles once. It is slow, but it never depends on remembering a rule.
Questions
How do I know which derivative rule to use?
Check in order: sum, constant multiple, product or quotient, chain, then standard form. Most expressions need two or three in sequence.
Are the answers on this page verified?
Yes. Every answer here is produced by the site's own engine when the page is built, and the engine's results are checked against numerical differentiation in the test suite.
Where can I get an endless supply of problems?
The practice card on the main calculator generates a fresh problem on demand, filtered to one kind of function — trigonometric, exponential and log, polynomial and root, inverse trigonometric, or hyperbolic.
What derivative problems are on Calculus 1 exams?
Almost exactly the four sets above: the rules applied singly, then products and quotients, then the chain rule, then mixed problems where recognising the rule is the test. Related rates and optimisation build on the same skills.
How do I check an answer I worked out by hand?
Type your function into any calculator on this site and compare, or open the problem directly — every problem below links into the calculator with the expression preloaded.