Derivative Calculator

Welcome to the Derivative Calculator

Calculate the derivative of your formula online with a detailed step-by-step solution you can follow. Just enter your mathematical function to see a live preview. Our calculator then walks you through the differentiation step by step — ideal for learning and checking your homework.

What is a derivative? 🤔

The derivative of a function is one of the most important concepts in calculus. Simply put, the derivative at a specific point describes the slope of the function’s graph exactly at that point. It tells us how much the function’s value changes when the input is varied slightly. With the derivative, denoted as $f'(x)$ or $\frac{d}{dx}f(x)$, we can analyze extrema, inflection points, and the overall monotonic behavior of functions.

How the derivative calculator works ⚙️

This calculator is more than just a tool for quickly finding results – it is designed as a learning aid to help you understand derivatives more easily.

Specifically developed for students and learners, it breaks down each function into its smallest logical parts and applies the appropriate differentiation rules. No steps are skipped. Even basic derivatives like $\frac{d}{dx}(x) = 1$ are shown explicitly, so the entire calculation process can be followed without gaps.

This way, you can dive into the often confusing topic of calculus even without deep prior knowledge and be well prepared for your next exam. 🎓

The most important differentiation rules 📚

To differentiate more complex functions, you break them into simpler parts and apply basic rules. Our calculator supports these and many more, showing every step in detail.

Power Rule

Used for powers like $x^n$. The exponent becomes a factor and is then reduced by one.

$\frac{d}{dx}\left(u^{c}\right)=c\,u^{c-1}\,u'$

Sum/Difference Rule

Allows differentiating sums and differences term by term. Each part is handled separately.

$\frac{d}{dx}(u \pm v \pm \dots) = \frac{d}{dx}(u) \pm \frac{d}{dx}(v) \pm \dots$

Product Rule

Needed when two functions $u(x) \cdot v(x)$ are multiplied. The formula is $(u \cdot v)' = u'v + uv'.

$\frac{d}{dx}(u \cdot v)=u'v+uv'$

Quotient Rule

The rule for differentiating quotients. Its structure is $\frac{u'v - uv'}{v^2}$.

$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u'v-uv'}{v^{2}}$

Chain Rule

The key rule for “nested” functions: differentiate the outer function and multiply by the derivative of the inner one.

$\frac{d}{dx}f(u)=f'(u)\cdot u'$

Exponential Rule

The derivative of the natural exponential $e^x$ is $e^x$ again. For a more complex exponent, the chain rule applies.

$\frac{d}{dx}(e^{u})=e^{u}\cdot u'$

Chain rule for the natural logarithm

The derivative of the natural logarithm $\ln(x)$ is $\frac{1}{x}$. For composed expressions, you need the chain rule.

$\frac{d}{dx}\big(\ln({u})\big) = \frac{u'}{{u}}$

Constant Factor Rule

A constant factor in front of a function is preserved during differentiation and multiplies the result.

$\frac{d}{dx}(c\,u)=c\,u'$