Derivative calculator that names the rule it used
Differentiate polynomials, trig, exponential and logarithmic expressions, in any combination. The chain, product and quotient rules are applied automatically, and each step says which rule it applied and to which part, because knowing the rule is what transfers to the next question.
Part of SolveStep's calculus solver. For the reverse operation, see the integral calculator.
Five rules cover almost every derivative you will meet
A derivative measures a rate of change: how fast the output moves when the input moves. Geometrically it is the slope of the tangent line at a point, which is why the slope of a straight line is the simplest derivative there is, the same everywhere.
Every derivative question after that is really a question about which rule applies, and the rules are decided by the shape of the expression rather than by what functions appear in it. A sum differentiates term by term. A product needs the product rule. A quotient needs the quotient rule. A function inside another function needs the chain rule. Recognising the shape is the skill; the arithmetic that follows is mechanical.
This calculator names the rule at each step rather than only showing the result, because the naming is the part that transfers to the next problem. Getting 6x + 2 back tells you nothing you can reuse; being told the power rule was applied term by term does.
Power rule
d/dx[xⁿ] = n·xⁿ⁻¹
Bring the exponent down in front, then subtract one from it. Works for every polynomial term.
Product rule
(fg)′ = f′g + fg′
Two things multiplied. Differentiate each in turn, leaving the other untouched, then add.
Quotient rule
(f/g)′ = (f′g − fg′)/g²
The order in the numerator matters, unlike the product rule, and reversing it flips the sign.
Chain rule
d/dx[f(g(x))] = f′(g(x))·g′(x)
A function inside a function. Differentiate the outside, leave the inside alone, multiply by the inside's derivative.
How to differentiate an expression, step by step
Differentiation distributes over addition and subtraction, so 3x² + 2x − 7 becomes three separate problems. Constants differentiate to zero, which is why the −7 simply disappears.
d/dx[5x³] = 5 · d/dx[x³] = 5 · 3x² = 15x². The constant carries straight through and never needs a rule of its own.
Bring the exponent down as a coefficient and reduce it by one. x⁵ becomes 5x⁴, and x becomes 1 because x = x¹ and x⁰ = 1.
Do not try to differentiate each factor separately and multiply. (x²)(x³) would give 2x · 3x² = 6x³, while the correct answer is 5x⁴. That mistake is the single most common one in the topic.
sin(3x + 1) is sin applied to 3x + 1. Differentiate the outside to get cos(3x + 1), then multiply by the derivative of the inside, which is 3.
One of each rule, written out
Power rule, term by term
d/dx(3x² + 2x − 7)- 3x²: bring the 2 down and reduce the power, giving 6x.
- 2x: the power is 1, so this gives 2.
- −7: a constant has no rate of change, so it differentiates to 0.
- Add the pieces back together.
Chain rule
d/dx(sin(3x + 1))- Identify the outside and the inside: sin is the outside, 3x + 1 is the inside.
- Differentiate the outside, leaving the inside exactly as it is: cos(3x + 1).
- Differentiate the inside on its own: d/dx(3x + 1) = 3.
- Multiply the two: 3cos(3x + 1). Forgetting that final multiplication is the classic chain-rule slip.
Quotient rule
d/dx(x² / (x + 1))- Set f = x² and g = x + 1, so f′ = 2x and g′ = 1.
- Apply (f′g − fg′)/g², keeping that order: (2x(x + 1) − x²(1)) / (x + 1)².
- Expand the numerator: 2x² + 2x − x² = x² + 2x.
- So the derivative is (x² + 2x)/(x + 1)². Swapping the two numerator terms would give the negative of the right answer.
Coverage, stated plainly
Polynomials of any degree, sin, cos, tan, e₁, a₁, ln, log, sqrt, cbrt and the inverse trig functions asin, acos and atan, combined by sums, products, quotients and composition to any depth. The chain, product and quotient rules are applied automatically and each is named in the steps as it is used.
Trig arguments are treated in radians, which is the standard convention in calculus and the only convention in which d/dx(sin x) = cos x is true. Logarithmic differentiation, for cases like x₀ where x appears in both the base and the exponent, is not supported yet and is reported as such rather than guessed at.
Derivatives, limits and integrals
A derivative is defined as a limit: the slope of the secant line as the two points merge. If you want to see that definition in action rather than take the rules on trust, the limit calculator evaluates the difference quotient directly.
Integration is the reverse operation, and the integral calculator handles the polynomial case with the constant of integration included. To see the tangent line a derivative describes, plot the function in the graphing calculator.
Derivative calculator FAQ
How do you find the derivative of a function?
Split it into terms, then match each term to a rule. A power gets the power rule: bring the exponent down and reduce it by one. Two things multiplied get the product rule, a quotient gets the quotient rule, and anything nested inside something else gets the chain rule. Constants differentiate to zero.
What is the power rule?
d/dx[xⁿ] = n·xⁿ⁻¹. Bring the exponent down as a coefficient and subtract one from it. So x⁵ differentiates to 5x⁴, and x differentiates to 1 because x = x¹ and x⁰ = 1. It applies term by term across a whole polynomial.
When do you use the chain rule?
Whenever one function sits inside another, such as sin(3x + 1), e^(2x) or (x² + 1)⁵. Differentiate the outer function leaving the inner one untouched, then multiply by the derivative of the inner function. Forgetting that final multiplication is the most common error in the topic.
What is the difference between the product rule and the quotient rule?
The product rule, (fg)′ = f′g + fg′, adds its two terms, so the order does not matter. The quotient rule, (f/g)′ = (f′g − fg′)/g², subtracts, so the order does matter and reversing it gives the negative of the correct answer.
Why can't you just differentiate each factor and multiply?
Because differentiation is not multiplicative. Take (x²)(x³) = x⁵, whose derivative is 5x⁴. Differentiating each factor and multiplying gives 2x · 3x² = 6x³, which is wrong. The product rule exists precisely to handle this.
Does this derivative calculator use radians or degrees?
Radians, which is the standard convention in calculus and the only one in which d/dx(sin x) = cos x holds. SolveStep's arithmetic mode uses degrees for plain sin() and cos() evaluation, since that is what school geometry expects, and the two modes are kept separate on purpose.
Is this derivative calculator free and does it show steps?
Yes to both. Every derivative comes with numbered steps that name the rule applied at each stage, there is no paid tier, no account, and nothing you type is sent to a server.
Tools that go with derivatives
Integral Calculator
The reverse operation, with the constant of integration included.
Limit Calculator
The limit that defines the derivative, worked step by step.
Calculus Solver
Derivatives and integrals together in one tool.
Graphing Calculator
Plot the function and see the tangent the derivative describes.
Trigonometry
The degrees-or-radians question, settled before you calculate.