Derivative Calculator

Type a function, click Calculate. This derivative calculator gives exact derivatives instantly, right on the page.

• Function to differentiate — the box comes pre-filled with x^3 + 2x. Replace it with any function, or tap an example button such as x^2, sin(x), e^x, or 1/x
• With respect to — pick the variable you are differentiating with respect to: x, y, or t. Almost every problem in a first calculus course uses x

Click Calculate and the derivative calculator returns the derivative: for the pre-filled $x^3 + 2x$ it gives $3x^2 + 2$ — the slope of the function at every point, computed exactly.

This page is more than a static derivative calculator — it is also an AI tutor you can ask questions. The question box under the form takes full questions, photos of the problem, and follow-ups:

• Ask in plain words. “What is the derivative of x^3 + 2x, one step at a time?” The AI reads it and differentiates it with every rule named.
• Take a photo of your homework. No typing: the AI reads the function and shows the full solution, step by step.
• Ask about the ideas, not just the answer. “Why does the chain rule need an extra factor?” The AI explains step by step — every result still comes from the derivative calculator on this page, so nothing is guessed.
• Work backwards, then keep asking. Give a derivative and ask which functions produce it. Follow-up questions are welcome: ask “why?” as often as you need — the derivative calculator page keeps teaching.

A speedometer does something algebra cannot: it reports your speed at one exact instant. Ordinary rates compare two moments — 100 km over 2 hours — but a derivative measures change at a single point. That is the first reading: a derivative is an instantaneous rate of change.

The second reading is geometric. Draw a curve and zoom into one point: the more you zoom, the straighter the curve looks, and the slope of that line is the derivative there — the slope of the tangent line. Ask “how steep?” anywhere, and the derivative answers.

Both readings share one notation:

$f'(x)=\dfrac{dy}{dx}$

The sign tells direction: $f'(x) > 0$ rising, $f'(x) < 0$ falling, $f'(x) = 0$ flat — a hilltop or a valley floor. Type any function into the derivative calculator above and read its steepness everywhere at once.

Every answer the derivative calculator produces is built from six short rules. Learn them once and you can differentiate almost anything by hand — and check it here.

The workhorse is the power rule:

$\dfrac{d}{dx}x^n=n\,x^{n-1}$

Bring the exponent down in front, then knock the exponent down by one. It is why $x^3$ differentiates to $3x^2$. Around it sit the supporting rules:

The last two deserve respect. The product rule handles two functions multiplied together. The chain rule handles functions inside functions — $\sin(2x)$ is sine with $2x$ nested inside, and Example 2 below works it through. When a derivative calculator answer looks longer than its input, one of these two rules is usually the reason.

Differentiate $x^3 + 2x$ — the function pre-filled in the derivative calculator above.

1. Split at the plus sign. The sum rule says differentiate each term on its own.
2. Power rule on $x^3$: bring the 3 down front and drop the exponent to 2, giving $3x^2$.
3. Rewrite $2x$ as $2x^1$: the power rule gives $2 \cdot 1 \cdot x^0 = 2$ — the coefficient simply survives.
4. Add the pieces back:

$\dfrac{d}{dx}\left(x^3+2x\right)=3x^2+2$

Notice the input has no plain number to erase. Add one — say $x^3 + 2x + 5$ — and the 5 contributes nothing: the derivative would still be $3x^2 + 2$, because constants differentiate to 0.

Click Calculate on the derivative calculator above and it confirms $3x^2 + 2$ instantly.

Two more differentiations, both one click away in the derivative calculator.

The trig family. Type sin(x) and the derivative calculator returns $\cos x$:

$\dfrac{d}{dx}(\sin x)=\cos x$

Sine and cosine trade places as they differentiate — the pattern every calculus student memorizes first.

The chain rule appears. Now type sin(2x). This is not sine alone: it is sine with $2x$ nested inside, so the chain rule applies:

1. Outer function first: the derivative of $\sin$ is $\cos$, evaluated at the inner function — $\cos(2x)$.
2. Times the inner derivative: the derivative of $2x$ is $2$.
3. Multiply:

$\dfrac{d}{dx}\left(\sin(2x)\right)=2\cos(2x)$

The factor of 2 is the whole point. Forget it and the answer is half right — literally. Type both functions into the derivative calculator above and watch the 2 appear the moment the inside stops being plain $x$.

1. Forgetting the inner derivative. The chain rule multiplies: $\dfrac{d}{dx}\sin(2x) = 2\cos(2x)$, not $\cos(2x)$. Any function with a function inside needs that extra factor — it is the most commonly lost point on calculus tests.
2. Panicking at negative exponents. The power rule does not stop at positive powers. Write $\dfrac{1}{x}$ as $x^{-1}$ and differentiate: $-x^{-2}$. The minus sign is correct — do not “fix” it. The derivative calculator handles negative exponents exactly as it handles positive ones.
3. Treating a constant like it differentiates to 1. Constants differentiate to 0, not 1. In $x^2 + 3x + 5$, the 5 vanishes: the derivative is $2x + 3$, with no lone 1 hanging on the end. Check any answer from the derivative calculator above — constants never survive.