What is a derivative?

A derivative measures how fast a function changes at any given point. Formally, the derivative of f(x) is defined as: f'(x) = lim(h→0) [f(x+h) - f(x)] / h This is the slope of the tangent line to the curve at point x. If f(x) gives position, f'(x) gives velocity. If f(x) gives velocity, f'(x) gives acceleration. Notations for derivatives: - f'(x) — Lagrange notation - dy/dx — Leibniz notation - Df(x) — operator notation All mean the same thing: the instantaneous rate of change of f with respect to x.

How do you apply the chain rule?

The chain rule is used when differentiating a composite function — a function inside another function. Chain rule: d/dx[f(g(x))] = f'(g(x)) * g'(x) In words: differentiate the outer function (leaving the inner alone), then multiply by the derivative of the inner function. Examples: - d/dx[sin(x^2)]: outer=sin, inner=x^2 = cos(x^2) * 2x = 2x*cos(x^2) - d/dx[(3x+1)^5]: outer=u^5, inner=3x+1 = 5*(3x+1)^4 * 3 = 15*(3x+1)^4 - d/dx[e^(sin x)]: outer=e^u, inner=sin x = e^(sin x) * cos x - d/dx[ln(x^2+1)]: outer=ln(u), inner=x^2+1 = 1/(x^2+1) * 2x = 2x/(x^2+1)

How do you apply the product rule?

Use the product rule when differentiating two functions multiplied together. Product rule: d/dx[f(x)*g(x)] = f'(x)*g(x) + f(x)*g'(x) Memory aid: 'derivative of first times second, plus first times derivative of second.' Examples: - d/dx[x * sin(x)]: f=x, g=sin(x) = 1*sin(x) + x*cos(x) = sin(x) + x*cos(x) - d/dx[x^2 * e^x]: f=x^2, g=e^x = 2x*e^x + x^2*e^x = e^x(2x + x^2) = x*e^x*(x+2) - d/dx[x * ln(x)]: f=x, g=ln(x) = 1*ln(x) + x*(1/x) = ln(x) + 1

What is the power rule and when does it apply?

The power rule is the most used differentiation rule: d/dx(x^n) = n * x^(n-1) It applies for ANY real exponent n — positive, negative, fractional, or zero. Examples: - d/dx(x^3) = 3x^2 - d/dx(x^10) = 10x^9 - d/dx(x) = 1x^0 = 1 - d/dx(x^0) = d/dx(1) = 0 - d/dx(x^(-1)) = d/dx(1/x) = -1*x^(-2) = -1/x^2 - d/dx(x^(1/2)) = d/dx(sqrt(x)) = (1/2)*x^(-1/2) = 1/(2*sqrt(x)) - d/dx(x^(-3)) = -3*x^(-4) = -3/x^4 For polynomials, apply the power rule to each term and use the sum rule: d/dx(3x^4 - 2x^2 + 7) = 12x^3 - 4x

Why is the derivative of e^x equal to e^x?

e^x is the unique function that equals its own derivative. This is what makes e the natural base for exponential functions. Proof from the limit definition: d/dx(e^x) = lim(h→0) [e^(x+h) - e^x] / h = lim(h→0) e^x * [e^h - 1] / h = e^x * lim(h→0) [e^h - 1] / h = e^x * 1 = e^x The key step uses the fact that lim(h→0) (e^h-1)/h = 1, which is essentially the definition of e. For other bases: d/dx(a^x) = a^x * ln(a) Since ln(e) = 1, d/dx(e^x) = e^x * 1 = e^x. Practical consequence: e^x models continuous growth/decay perfectly because its rate of change always equals its current value — the growth rate is always proportional to the current amount.

Derivative Calculator — Step-by-Step Symbolic Differentiation

The Derivative Calculator computes exact symbolic derivatives for any function with full step-by-step working. Supports all standard calculus functions: polynomials (power rule), trig (sin, cos, tan), inverse trig (arcsin, arccos, arctan), exponentials (e^x), natural logarithm (ln x), and square roo...

DERIVATIVE CALCULATOR

f(x) = (with respect to x)

Use * for multiplication, ^ for powers. Press Enter or click d/dx.

QUICK EXAMPLES

Enter a function above and press d/dx or Enter

DIFFERENTIATION RULES

Constant ruled/dx(c) = 0c is any constant

Variable ruled/dx(x) = 1

Power ruled/dx(xⁿ) = n·xⁿ⁻¹n is any real number

Constant multipled/dx(c·f) = c·f′

Sum ruled/dx(f+g) = f′+g′

Difference ruled/dx(f−g) = f′−g′

Product ruled/dx(f·g) = f′g+fg′

Quotient ruled/dx(f/g) = (f′g−fg′)/g²

Chain ruled/dx(f(g)) = f′(g)·g′

sind/dx(sin x) = cos x

cosd/dx(cos x) = −sin x

tand/dx(tan x) = sec²x

eˣd/dx(eˣ) = eˣ

aˣd/dx(aˣ) = aˣ·ln(a)

ln xd/dx(ln x) = 1/x

√xd/dx(√x) = 1/(2√x)

arcsind/dx(arcsin x) = 1/√(1−x²)

arctand/dx(arctan x) = 1/(1+x²)

INPUT SYNTAX

x^3x³ (power)

2*x2x (multiplication)

sin(x)sine of x

exp(x)eˣ

ln(x)natural log

sqrt(x)√x

arctan(x)tan⁻¹(x)

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DIFFERENTIATION RULES REFERENCE

DERIVATIVE RULES

RULE	FORMULA	DESCRIPTION
Constant rule	d/dx(c) = 0	Derivative of any constant is zero
Power rule	d/dx(x^n) = n*x^(n-1)	Applies for any real exponent n
Sum rule	d/dx(f+g) = f'+g'	Differentiate term by term
Product rule	d/dx(f*g) = f'g + fg'	For products of two functions
Quotient rule	d/dx(f/g) = (f'g - fg') / g^2	For ratios of two functions
Chain rule	d/dx(f(g(x))) = f'(g(x)) * g'(x)	For composed functions
d/dx(sin x)	cos x
d/dx(cos x)	-sin x
d/dx(tan x)	sec^2(x)
d/dx(e^x)	e^x	Unique: derivative equals itself
d/dx(ln x)	1/x
d/dx(arctan x)	1/(1+x^2)
d/dx(arcsin x)	1/sqrt(1-x^2)

HOW TO USE

1
Type your function in the input box using standard notation: x^3 for x³, 2*x for 2x, sin(x) for sine, exp(x) for eˣ, ln(x) for natural log, sqrt(x) for square root. Press Enter or click d/dx.
2
The derivative appears instantly in the result box as f′(x). The step-by-step panel below shows which differentiation rule was applied at each stage — constant rule, power rule, chain rule, product rule, or quotient rule.
3
Use the quick example buttons to try common functions: x³, sin(x), cos(x²) (chain rule), eˣ, x·sin(x) (product rule), 1/x (quotient rule), arctan(x) (inverse trig).
4
The right panel shows the complete differentiation rules reference — all rules and trig derivatives in one place for quick lookup while studying.
5
The calculator supports all standard calculus functions: polynomials, trig (sin, cos, tan, sec, csc, cot), inverse trig (arcsin, arccos, arctan), exponentials (exp), logarithms (ln), and square roots (sqrt).

WORKED EXAMPLE

d/dx(x^3 + 2*x^2 - 5*x + 7): Sum rule applied term by term. d/dx(x^3)=3x^2 (power rule). d/dx(2*x^2)=4x (constant multiple + power rule). d/dx(-5*x)=-5 (constant multiple). d/dx(7)=0 (constant). Result: 3x^2 + 4x - 5. Chain rule example: d/dx(sin(x^2)) = cos(x^2) * 2x = 2x*cos(x^2).

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Last updated: April 29, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.