Derivative Calculator

Differentiate a function or implicit equation. Follow each rule, explore the curve and inspect regular, vertical or singular tangent points.

See the derivative, understand the rule

Differentiate a function or an implicit equation. Follow its parts, then inspect a point on the curve and its tangent.

f(x)Use ^ for powers, parentheses and x. ln is natural log; log is base 10. Trigonometric angles are in radians.

Derivative orderThe graph compares f with this derivative; the tangent always uses f′.

Inspect at xOptional. A number or constant expression, such as pi/2.

First derivative

6 \cdot \cos (3 \cdot x^{2}) \cdot x

The formula applies within the original function’s real domain.

At your point

f(1) ≈ 0.141120008; f′(1) ≈ -5.93995498

Tangent: y ≈ 0.141120008 + (-5.93995498)(x − (1)).

Curve and tangent

FunctionSelected derivativeTangent

Click near the curve to inspect a point. You can also enter coordinates using the keyboard. Plots are numerical illustrations; small features may be missed.

Plot window

Minimum coordinateBoth axes use the same scale.

Maximum coordinateAdjust to explore a wider part of the curve.

Follow the working

Choose a part to see its rule, the expression it acts on and the result. Inner parts are shown before the operations that use them.

Chain: sin

Differentiate this part with respect to x.

\sin (3 \cdot x^{2})

Its contribution becomes

6 \cdot \cos (3 \cdot x^{2}) \cdot x

Differentiate the outer function, then multiply by the derivative of its inner expression.

Supported expressions and accuracy

Real elementary expressions with x (and y in implicit mode): +, −, ×, ÷, ^, sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, exp, ln, log, sqrt, cbrt and abs; constants pi and e. Use one expression or one equation. Functions have one argument. Unsupported syntax is rejected.

Symbolic rational coefficients retain their exact values. Rational coordinate expressions retain exact equation membership; point values and plots are numerical approximations. Overflow or underflow can prevent numerical evaluation even when a symbolic formula is valid. A formula that cannot be evaluated may need a limit; this tool does not solve general singular points or certify all branches. An original-domain exclusion stays excluded. Zero to a nonpositive power is unsupported. Negative bases with exact odd-denominator rational powers use real roots.

Inputs are limited to 600 characters per expression, 160 parts and 24 nested levels. Large derivative expansions and walkthroughs exceeding 500 steps are stopped with an explanation. Numerical tangent-pole and finite-range guards may prevent a point calculation near a singularity.