f'Calculus · Differentiation
Derivative Calculator with every rule named
Type a function and get its derivative with every rule named at the step it's used. Differentiate — or "find dy/dx" — for free, with a graph to check it.
- Free steps
- Graph
- Checked answers
Try it
Shrink h. The secant becomes the tangent.
The derivative is the limit of secant slopes (f(x + h) − f(x))/h as h → 0. The lower strip is f′ itself: the slope at every point, drawn as a new function.
Slope negative — the curve is falling.
Why this derivative calculator is different
01
Every rule, with u, v, u′ and v′
Each step names its rule — power, product, quotient, chain — and writes out u, v, u' and v' before combining them, exactly as your teacher marks it. Constants such as ln 2 stay exact, never 0.693….
02
Checked two independent ways
First derivatives are compared with the numerical slope of f and re-derived by a second algebra engine; higher derivatives with 80-digit finite differences. Only then does the Verified stamp appear.
03
Curves and points too
Type x^2 + y^2 = 25 for implicit differentiation (dy/dx = -x/y), or add a point for the gradient, tangent and normal.
What is a derivative calculator?
A derivative calculator finds the derivative of a function — the formula for its rate of change — and shows how the answer is built. Geometrically, f'(a) is the gradient (slope) of the tangent line to y = f(x) at x = a: positive where the curve rises, negative where it falls and zero at a turning point.
Formally the derivative is a limit, f'(x) = lim _h → 0 (f(x+h) - f(x))/(h): the slope of a secant line through two points of the curve as the points merge (try it in the demo above). In practice you combine a few rules — power, product, quotient, chain — with a table of standard derivatives, and that is what this derivative calculator does, one named step at a time.
Derivatives give stationary points, rates of change (velocity is the derivative of position) and marginal cost, and fill the differentiation chapters of A-level, CBSE Class 11–12, AP Calculus and IB Mathematics.
How to use the derivative calculator
- 1Type a function such as `x^2 sin(x)` or `ln(x)/x`. You can write `sin x`, `sqrt(x)`, `e^x`, `ln`, `log` (base 10), `log_2(x)`, `pi`, `∛x` or paste LaTeX like `\frac{1}{x}`; the preview shows how it was read.
- 2For dy/dx of a curve, type the whole equation, e.g. `x^2 + y^2 = 25` or `sin(xy) = x` — the calculator switches to implicit differentiation.
- 3Pick the order (f', f'' or f'''), or write it in the problem: `d^2/dx^2 x e^x`. Add a point in At x = (`2`, or `(3, 4)` on a curve), or type `x^3 - 3x at x = 2`, to get the gradient, the tangent and the normal.
- 4Read the steps — One step at a time lets you try each line first — and open Graph to see f and f' together: where f' is zero, f turns.
Differentiation rules, step by step
Almost every derivative at school is built from these rules. The calculator names the one it uses at each step:
Power rule
Bring the power down and subtract one — also for negative and fractional powers: √(x) = x^(1/2) gives (1)/(2√(x)).
Product rule
For two functions multiplied together. Name u and v, find u' and v', then multiply crosswise and add — for x^2 sin x that is 2x sin x + x^2 cos x.
Quotient rule
For one function divided by another. The top-derivative term comes first, and the bottom stays squared.
Chain rule
For a function inside another: differentiate the outside, keep the inside, multiply by the derivative of the inside. d/dx cos (5x^2) = -10x sin (5x^2).
Exponentials, logarithms and trig
e^x is its own derivative; any other base brings out its natural logarithm, kept exact as ln 2. For x^x, use logarithmic differentiation.
Implicit differentiation
When y is not written as a function of x, differentiate both sides, multiply every y-term by dy/dx (chain rule), collect those terms and solve. At (3, 4) the slope is -3/4.
Derivative calculator worked example: x² sin x
- 01
The function is a product, so use the product rule with u = x^2 and v = sin x.
(uv)' = u'v + uv' - 02
Differentiate each factor: u' = 2x by the power rule and v' = cos x from the standard derivatives.
- 03
Multiply crosswise and add.
d/dx[x^2 sin x] = 2x sin x + x^2 cos x
Mistakes the derivative calculator catches
Forgetting the chain rule
The derivative of sin (3x) is 3 cos (3x), not cos (3x). Whenever the inside is not just x, multiply by the derivative of the inside — the steps show that factor as u'.
Multiplying derivatives in a product
(uv)' is not u'v'. For x^2 sin x the answer is 2x sin x + x^2 cos x, not 2x cos x.
Swapping the quotient rule
The numerator is u'v - uv', top-derivative first. Reversing the terms flips the sign of the whole answer.
Dropping dy/dx on a y-term
In implicit differentiation d/dx[y^2] = 2ydy/dx, not 2y. Forgetting the factor makes dy/dx impossible to solve for.
Derivative calculator FAQ
Is this derivative calculator free?+
Yes. Typed derivatives are free: every step, the graph, implicit differentiation, the tangent at a point and the practice problems, with no sign-up and no adverts. Photo questions are read by AI and have a daily allowance, but steps for typed problems are never locked.
Does the derivative calculator show steps?+
Yes. Each step names the rule it uses — power, product, quotient, chain, exponential, logarithmic or trigonometric — and writes out u, v, u' and v' before combining them. You can read all the steps at once, or one at a time so you can try each line before you see it.
How do I find dy/dx by implicit differentiation?+
Type the whole equation, for example `x^2 + y^2 = 25` or `x^3 + y^3 = 6xy`. The calculator differentiates both sides with respect to x, adds the factor dy/dx to every y-term, collects those terms and solves. Add a point such as `(3, 4)` to get the slope and tangent there; choose f'' for (d^2y)/(dx^2).
How do I find the second derivative?+
Set Order to f'' (or f'''), or start the problem with `d^2/dx^2`. The calculator differentiates step by step, shows each derivative in turn and checks the last one against 80-digit finite differences of the original function — a test that does not use the steps at all.
Can it find the gradient, tangent line and normal at a point?+
Yes. Enter a value in At x = or type `at x = 2` after the function. You get f'(a) as an exact value where possible (such as (√(2))/(2)), the tangent y - f(a) = f'(a)(x - a), the normal with gradient -(1)/(f'(a)), and the tangent drawn on the graph.
How do you know the derivative is correct?+
Two checks that do not rely on the steps: the answer is compared with the numerical slope of the original function at up to 20 points, and re-derived by a second computer-algebra engine. For curves, slopes are measured along the curve itself. If a check fails you see a warning, not a stamp.
Why does the answer keep ln 2 instead of a decimal?+
Because ln 2 is exact and 0.693 ldots is not. Exam answers expect 2^x ln 2 or (1)/(x ln 10), and rounding early spreads errors into every later step and can cost marks. The decimal value still appears, separately, when you evaluate at a point.
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