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Derivative Calculator

Calculate the symbolic derivative of a function, nth derivative and numerical value f'(a) with graph plot.

f(x) =
x² x³−2x+1 sin(x) cos(x) eˣ ln(x) √x x²·sin(x) sin(x)/x (x²+1)/(x−1) x·eˣ arctan(x)
f(x) =
x⁴ sin(x) eˣ x⁵−3x³+x cos(x)
f(x) =
x²+sin(x), a=1 eˣ, a=0 ln(x), a=1 x³, a=2 sin·cos, a=0
↵ Enter to differentiate
Enter a function and click Differentiate.

How to calculate a derivative online?

The derivative of a function f(x) represents the instantaneous rate of change of the function at each point. Written f'(x) or df/dx, it is fundamental in mathematical analysis, physics and economics. Our calculator uses symbolic differentiation to produce the exact derivative expression.

Main differentiation rules

The fundamental rules are: sum rule (f+g)' = f'+g', product rule (f·g)' = f'·g + f·g', quotient rule (f/g)' = (f'g−fg')/g², and chain rule (f∘g)' = f'(g(x))·g'(x). Standard derivatives: (xⁿ)' = n·xⁿ⁻¹, (sin x)' = cos x, (cos x)' = −sin x, (eˣ)' = eˣ, (ln x)' = 1/x.

Second derivative and interpretation

The second derivative f''(x) gives the curvature of the function. If f''(x) > 0, the function is convex (curving upward); if f''(x) < 0, it is concave. Inflection points are at zeros of f''(x). The second derivative is also acceleration in physics (derivative of velocity).

Frequently asked questions

The calculator supports all standard functions: polynomials (x^2, x^3+2x), trigonometric functions (sin, cos, tan, asin, acos, atan), exponentials (exp, e^x), l... The calculator supports all standard functions: polynomials (x^2, x^3+2x), trigonometric functions (sin, cos, tan, asin, acos, atan), exponentials (exp, e^x), logarithms (log for ln), square root (sqrt), and all their combinations (sum, product, quotient, composition). Symbolic differentiation is powered by the math.js library.

The nth derivative consists of differentiating the function n times successively. The second derivative f''(x) is the derivative of f'(x), the third derivative ... The nth derivative consists of differentiating the function n times successively. The second derivative f''(x) is the derivative of f'(x), the third derivative f'''(x) is the derivative of f''(x), etc. Our tool calculates successive derivatives up to order 4 and displays the result at each step. Example: for f(x) = x^4, we get f'=4x³, f''=12x², f'''=24x, f''''=24.

The value f'(a) is the value of the derivative function at point x = a. It represents the slope of the curve f(x) at that point, i.e., the inclination of the ta... The value f'(a) is the value of the derivative function at point x = a. It represents the slope of the curve f(x) at that point, i.e., the inclination of the tangent to the curve at point (a, f(a)). If f'(a) > 0, the function is increasing at a; if f'(a) < 0, it is decreasing; if f'(a) = 0, the point may be an extremum (local maximum or minimum).

The chain rule applies when differentiating a composite function f(g(x)). The formula is: (f∘g)'(x) = f'(g(x)) × g'(x). Example: for sin(x²), set f(u) = sin(u) ... The chain rule applies when differentiating a composite function f(g(x)). The formula is: (f∘g)'(x) = f'(g(x)) × g'(x). Example: for sin(x²), set f(u) = sin(u) and g(x) = x². Then f'(u) = cos(u) and g'(x) = 2x. The derivative is: cos(x²) × 2x = 2x·cos(x²). This rule is automatically applied by our calculator.

The graph simultaneously shows f(x) in purple and f'(x) in red. Where f'(x) = 0 (red curve crosses the axis), f(x) reaches a local maximum or minimum. Where f'(... The graph simultaneously shows f(x) in purple and f'(x) in red. Where f'(x) = 0 (red curve crosses the axis), f(x) reaches a local maximum or minimum. Where f'(x) > 0, f(x) is increasing (going up); where f'(x) < 0, f(x) is decreasing (going down). Sign changes of f'(x) correspond to extrema of f(x).

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