Derivative Calculator
Differentiate an expression symbolically with the product, quotient and chain rules applied and named at each step, plus higher derivatives and evaluation at a point.
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Differentiation is mechanical, which is why it can be automated
Every derivative follows from a small set of rules applied recursively: the derivative of a sum is the sum of the derivatives, the product rule handles multiplication, the quotient rule division, and the chain rule composition. There is no creativity required, unlike integration, which is why symbolic differentiation is a solved problem and symbolic integration is not. Working through the rules by hand is where errors creep in, almost always in a chain rule step that was skipped.
The chain rule is where mistakes happen
The derivative of sin of x squared is not cosine of x squared. It is cosine of x squared multiplied by the derivative of the inside, which is 2x. Every composed function needs that inner factor, and forgetting it is the single most common differentiation error. Each application of the chain rule is named explicitly in the steps below so the inner derivative is visible rather than assumed.
Simplification is a separate problem
The raw output of applying the rules is correct and frequently ugly, containing terms multiplied by one and added to zero. Simplification cleans those up, and beyond the obvious cases deciding which of two equivalent forms is simpler is a matter of judgement rather than a rule. The result here is simplified conservatively, so it is always correct and occasionally longer than the form you would write by hand.
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Frequently Asked Questions
Which functions are supported?
Polynomials, sums, products, quotients and powers, together with sin, cos, tan, exp, ln, log, sqrt, the inverse trigonometric functions and the hyperbolic ones. Composition of any of these is handled by the chain rule.
Why is my answer written differently?
Because there are many equivalent forms of the same derivative and choosing the tidiest is judgement rather than a rule. Simplification here is conservative, so the result is always correct and is sometimes longer than the form you would write by hand.
What is the most common differentiation mistake?
Omitting the chain rule factor. The derivative of sin of x squared is cosine of x squared times 2x, not just cosine of x squared. Each chain rule application is named in the steps so the inner derivative is visible.
Can it do higher derivatives?
Yes. The second and third derivatives are computed by differentiating the result again. Expressions grow quickly with each pass, which is why the third derivative of a quotient is usually much longer than the first.
Why is integration harder than differentiation?
Because differentiation is a finite set of rules applied mechanically to any expression, while integration has no such complete procedure. Many elementary functions have antiderivatives that cannot be written in elementary terms at all.
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How to Use
Enter a function of x to see its derivative and the rules used to obtain it.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.
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