Derivative Calculator

Type sin, sqrt, ^, / naturally. Shorthand like sin3x works; use * for products.
Preview type a function above…
Examples by rule
Power
Product
Quotient
Chain
Trig / log
Functions: sin, cos, tan, ln (or log), sqrt, e^x.
Arc trig: arcsin(x), arccos(x), arctan(x).
Operators: use * for multiplication, / for division, ^ for powers.
Shorthand: sin3x is accepted (same as sin(3*x)).
Leave "Evaluate at" blank for a symbolic derivative; fill it to also get f(n)(a).
d/dx

Ready when you are

Type a function above and hit Differentiate.

Differentiate a function to see its graph.

Template:
Power rule d/dx[xn] = n·xn−1
Product rule (fg)′ = f′g + fg′
Chain rule d/dx[f(g(x))] = f′(g)·g′(x)

Frequently asked

A derivative measures the instantaneous rate of change of a function. Geometrically, f'(a) equals the slope of the tangent line to y = f(x) at the point (a, f(a)). The derivative is defined as f'(x) = limh→0 [f(x+h) − f(x)] / h. Derivatives are fundamental to calculus, physics, engineering, and optimization.
(1) Identify the function type — polynomial, trig, exponential, or composite. (2) Apply the matching rule: power rule for x^n, product rule for fg, quotient rule for f/g, or chain rule for f(g(x)). (3) Simplify the result. (4) Verify by checking at specific points.
The chain rule states d/dx[f(g(x))] = f'(g(x)) · g'(x). Use it when differentiating composite functions — a function inside another function. For example, d/dx[sin(x²)] = cos(x²) · 2x. The outer function is sin and the inner is x². Most used rule in calculus.
The first derivative f'(x) gives the rate of change and slope of the tangent line. The second derivative f''(x) measures how the rate of change itself is changing — it determines concavity. If f''(x) > 0 the graph is concave up; if < 0 concave down. In physics, if f is position, f' is velocity and f'' is acceleration.
(1) Compute f'(x). (2) Set f'(x) = 0 and solve for x. (3) Also check where f'(x) is undefined. These x-values are critical points. Use the second-derivative test: if f''(c) > 0 it's a local min, if < 0 a local max, if = 0 the test is inconclusive.
The five basics: Power d/dx[xn] = nxn−1, Product d/dx[fg] = f'g + fg', Quotient d/dx[f/g] = (f'g − fg')/g², Chain d/dx[f(g)] = f'(g)g', Sum d/dx[f+g] = f' + g'. Most derivatives are combinations of these.
Yes. Use the order toggle at the top to compute 1st, 2nd, 3rd, 4th, or 5th derivatives. For f(x) = x4, f'' is 12x², f''' is 24x, f'''' is 24. Each order shows full step-by-step solutions. The second derivative reveals concavity; higher derivatives show up in Taylor series and physics (jerk, snap).
Implicit differentiation finds dy/dx when y is defined implicitly by an equation like x²+y²=25. Differentiate both sides with respect to x, treating y as a function of x. This calculator differentiates expressions with respect to any variable (x, y, t, …). Enter expressions involving multiple variables like x²y or sin(x)cos(y) and select the variable. The built-in Python compiler supports full implicit differentiation via SymPy.