Derivative Calculator

The derivative of a function measures how fast its output changes as its input changes, at a single instant rather than over an interval. On a graph it is the slope of the tangent line; in a word problem it is a rate, such as feet per second or dollars per unit. This page explains the rules the calculator applies, works through examples with real numbers, and covers all three tabs: ordinary derivatives up to the fifth order, partial derivatives of functions of x, y and z, and implicit differentiation of equations.

Three sister tools cover the rest of introductory calculus. The Integral Calculator runs the process in reverse and finds antiderivatives and areas. The Limit Calculator evaluates the kind of limit that the derivative is built on, and the Graphing Calculator draws the function so you can see the tangent line this page computes.

What a derivative measures

Picture a ball dropped from a bridge. Ignoring air resistance and using the textbook value of 16 feet per second squared for half of gravity's pull, it falls s(t) = 16t² feet in t seconds. Over the first two seconds it covers 64 feet, an average speed of 32 feet per second, but at the moment t = 2 it is moving faster than that average. The derivative captures the speed at that instant: s′(t) = 32t, so s′(2) = 64 feet per second. A derivative always carries output units per input unit, which makes the units the quickest sanity check on any answer.

On a graph, the derivative is the slope of the line that just touches the curve at one point. Where the derivative is positive the function rises, where it is negative the function falls, and where it equals zero the tangent line is horizontal, which is where maximums and minimums can occur. The slope between two separate points is the job of the Slope Calculator and the Rate of Change Calculator; the derivative is the value that slope settles on when the two points slide together into one.

The limit definition and the difference quotient

Every rule on this page follows from a single definition, the one given as equation 1.4.4 in the NIST Digital Library of Mathematical Functions:

The fraction inside the limit is the difference quotient: the slope of the secant line through the points (x, f(x)) and (x + h, f(x + h)). For f(x) = x² it simplifies cleanly:

As h shrinks to zero, 2x + h becomes 2x, and that is the derivative. The calculator performs this simplification for any polynomial you enter: for 3x² + 5x − 4 it returns 6x + 3h + 5, and dropping the h term leaves f′(x) = 6x + 5. For square roots, exponentials and sines the quotient does not collapse into a polynomial, but the numbers still converge. The table evaluates the difference quotient at x = 1 for four functions, and every row creeps toward the exact derivative in the last column.

Each step cuts h by a factor of 10, and for these smooth functions the gap to the exact value shrinks by roughly the same factor. That pattern is why the limit definition works, and also why it is a clumsy way to compute derivatives in practice: accuracy demands a tiny h, and tiny differences lose digits to rounding. The rules below give exact answers instead.

The differentiation rules at a glance

The calculator reads the structure of your formula and applies one of these rules to each piece. The derivative formulas for the exponential, logarithm, power and trigonometric functions are the ones listed in NIST DLMF sections 4.7, 4.20 and 4.24, and every example in the right-hand column is the calculator's own output.

A constant on its own differentiates to 0, which is why the −7 in x³ − 5x² + 4x − 7 disappears from its derivative, 3x² − 10x + 4. The power rule covers roots and reciprocals too, once you write them as powers: √x = x^1/2 has derivative 1/(2√x), and 1/x = x⁻¹ has derivative −1/x².

Worked example 1: the product rule and the chain rule

Differentiate f(x) = x² sin(3x), the default example in the calculator. The function is a product, so the product rule comes first, with u = x² and v = sin(3x):

The factor sin(3x) is itself a composition, an outer sine applied to the inner function 3x, so it needs the chain rule: the derivative of the outside is cos(3x) and the derivative of the inside is 3. Multiplying gives (sin 3x)′ = 3cos(3x), and therefore

At x = 1, in radians, sin 3 = 0.14112 and cos 3 = −0.98999, so f(1) = 0.14112 and f′(1) = 2(0.14112) + 3(−0.98999) = 0.28224 − 2.96998 = −2.68774. The tangent line passes through (1, 0.14112) with slope −2.68774, which puts its y-intercept at 0.14112 + 2.68774 = 2.82886. The calculator reports y = −2.68774x + 2.82886, and its limit check shows the difference quotient moving from −3.31992 at h = 0.1 to −2.68838 at h = 0.0001, closing in on the exact slope.

Worked example 2: the quotient rule and an exact tangent line

For f(x) = (x² + 1)/(x − 1), let u = x² + 1 and v = x − 1, so u′ = 2x and v′ = 1. The quotient rule gives

At x = 3 the function value is (9 + 1)/2 = 5 and the slope is (9 − 6 − 1)/4 = 1/2. The point-slope form y − 5 = ½(x − 3) rearranges to y = x/2 + 7/2, and the calculator returns exactly that line, in fractions rather than decimals, because every value along the way is rational. Choosing order 3 adds f″(x) = 4/(x − 1)³ and f‴(x) = −12/(x − 1)⁴, which give f″(3) = 1/2 and f‴(3) = −3/4.

The same derivative locates the turning points. The slope is zero when x² − 2x − 1 = 0, which happens at x = 1 ± √2. At x = 1 + √2 ≈ 2.41421 the function has a local minimum of 2 + 2√2 ≈ 4.82843, and the second derivative there is positive (about 1.41421), confirming the minimum. At x = 1 − √2 ≈ −0.41421 it has a local maximum of 2 − 2√2 ≈ −0.82843. The maximum sits below the minimum because the vertical asymptote at x = 1 separates the two branches of the graph.

Higher-order derivatives: position, velocity and acceleration

Differentiating a derivative gives the second derivative, then the third, and so on. In motion problems each one has a name. Suppose a cart's position along a track is s(t) = t³ − 6t² + 9t + 2 feet after t seconds. Its velocity is v(t) = s′(t) = 3t² − 12t + 9 feet per second, its acceleration is a(t) = s″(t) = 6t − 12 feet per second squared, and the third derivative, called jerk, is the constant 6. The calculator finds all of them in one pass: enter the function with t as the variable and choose order 3.

The velocity is zero at t = 1 and t = 3, so the cart stops and reverses at those moments. At t = 1 the acceleration is −6, which makes the position a local maximum of 6 feet; at t = 3 it is +6, a local minimum of 2 feet. The acceleration crosses zero at t = 2, where the backward speed peaks at 3 feet per second. That is the inflection point of s(t), where the graph switches from bending down to bending up.

Partial derivatives of functions of several variables

When a function depends on more than one variable, a partial derivative measures the change in one direction while the other variables stay fixed. Hold y constant and differentiate f(x, y) = x²y + 3xy³ with respect to x: the term x²y behaves like a constant times x², and 3xy³ like a constant times x. Doing the same with x held constant gives the partial derivative with respect to y:

At the point (1, 2), f = 26, ∂f/∂x = 2(1)(2) + 3(8) = 28 and ∂f/∂y = 1 + 9(1)(4) = 37. Together they form the gradient vector (28, 37), which points in the direction of steepest increase; its length, about 46.40, is the steepest slope available at that point. Second-order partials follow the same pattern. Differentiating ∂f/∂x with respect to y gives the mixed partial f_xy = 2x + 9y², and differentiating in the opposite order gives the same result, as Clairaut's theorem guarantees for functions whose second partials are continuous (DLMF equation 1.5.6). At (1, 2) both mixed partials equal 38, while f_xx = 2y = 4 and f_yy = 18xy = 36.

Implicit differentiation step by step

Some curves are easier to describe with an equation than with y = f(x). The circle x² + y² = 25 is the standard case: solving for y splits it into two functions, y = ±√(25 − x²). Implicit differentiation avoids the split. Differentiate both sides with respect to x and remember that y depends on x, so every y term picks up a factor of dy/dx from the chain rule:

At (3, 4) the slope is −3/4 and the tangent line is y = −3x/4 + 25/4. At (5, 0) the denominator is zero, which signals a vertical tangent, and the calculator says so instead of printing a division by zero. The general shortcut, which comes from the implicit function theorem in DLMF section 1.5, is dy/dx = −F_x/F_y after rewriting the equation as F(x, y) = 0. For the folium of Descartes, x³ + y³ = 6xy, F_x = 3x² − 6y and F_y = 3y² − 6x, so dy/dx = (2y − x²)/(y² − 2x) once the common factor of 3 cancels. The point (3, 3) lies on the curve because 27 + 27 = 54 = 6·3·3, its slope is (6 − 9)/(9 − 6) = −1, and its tangent line is y = −x + 6.

How to use this calculator

Pick a tab first. Derivative handles y = f(x) at orders 1 to 5, Partial handles functions of x, y and z, and Implicit takes an equation with an equals sign. Type the function the way you would write it on paper: 2x and x y both mean multiplication, ^ marks a power, and sin x, sin(x) and sinx all mean the sine of x. Use sqrt() for square roots, |x| or abs() for absolute values, ln for the natural logarithm, log for base 10, and e and pi for the constants. Superscripts pasted from a document, such as x², are converted automatically, and sin^2(x) is read as the square of sin(x). Numbers can be decimals or scientific notation such as 2.5e-3; write them without thousands separators, and put an operator between two numbers, because the calculator asks rather than guesses whether x2 means x^2 or 2x.

The three tabs make this a derivative calculator, a partial derivative calculator and an implicit differentiation calculator in one, and the step list works as a chain rule calculator: every time the chain rule is used it names the outer function, the inner function u and its derivative u′.

The result updates as you type. The headline is the derivative in readable math notation, with your original function underneath so you can confirm the calculator read the input the way you intended, followed by a plain-text version that pastes cleanly into other software. Enter a point to get the function value, the slope and the tangent line, or leave the box empty to skip it. The step list names each rule, the parts it was applied to and the derivative of every part. If the function uses a single letter other than x, such as t, the Derivative tab differentiates with respect to that letter; extra letters such as a and b are treated as constants, so ax² + bx + c returns 2ax + b.

Who uses a derivative calculator

Students in AP Calculus AB and BC and in college Calculus I use it to check homework, and the step list shows the intermediate work a grader expects. Physics students turn position functions into velocity and acceleration. In economics, the derivative of a cost function is the marginal cost: if producing q units costs C(q) = 0.02q² + 15q + 500 dollars, then C′(q) = 0.04q + 15, so at a production level of 100 units each extra unit costs about $19. The exact cost of going from 100 to 101 units is C(101) − C(100) = $19.02, which shows how closely the derivative tracks a one-unit change. (The calculator writes 0.04q as the exact fraction q/25.) Engineers use derivatives for flow and cooling rates, and partial derivatives drive the gradient methods that fit models in statistics and machine learning.

Key terms

• Derivative: f′(x) or dy/dx, the instantaneous rate of change; the slope of the tangent line.
• Differentiable: having a derivative at a point, which means the graph is smooth there, with no corner, cusp or jump.
• Tangent line: the line through (a, f(a)) with slope f′(a), written y = f(a) + f′(a)(x − a).
• Difference quotient: [f(x + h) − f(x)]/h, the secant slope whose limit is the derivative.
• Critical point: a point where f′(x) = 0 or f′ does not exist, the candidates for maximums and minimums.
• Higher-order derivative: f″, f‴ and beyond, found by differentiating repeatedly.
• Partial derivative: ∂f/∂x, the rate of change in one variable with the others held constant.
• Gradient: the vector of all first partial derivatives, written ∇f.
• Implicit differentiation: differentiating an equation in x and y without solving for y first.
• Chain rule: the rule for a function inside a function, (f(g(x)))′ = f′(g(x))·g′(x).

Why your answer may look different

A correct derivative can be written in several equivalent forms, and the calculator follows one consistent style: it multiplies out where that shortens the result, combines like terms, pulls out common factors such as e^x, and puts quotients over a common denominator. Three things commonly make its answer look unlike a textbook's even when both are right.

• Trigonometric identities are not applied. The derivative of ln(cos x) appears as −sin(x)/cos(x) rather than −tan(x), and the derivative of sin²(x) as 2sin(x)cos(x) rather than sin(2x).
• Factored versus expanded. The derivative of x²/(x + 3) is shown as (x² + 6x)/(x + 3)², while a textbook may write x(x + 6)/(x + 3)².
• Exact fractions. Decimals you type become fractions, so 0.5x² differentiates to x rather than 1.0x, and 0.04q appears as q/25.

The quickest check is numeric: evaluate both forms at two or three points. If the values agree, the answers agree.

Limitations

The calculator differentiates formulas, not graphs or data tables, and the formula it returns is valid wherever the original function is differentiable. It does not flag the points where that fails, so it gives x/|x| as the derivative of |x| even though no derivative exists at x = 0, and 2/(3∛x) for x^(2/3), the cube root of x², which has a cusp at the origin. When you evaluate at a point, it does report the value as undefined there, and it gives no slope at a point where the function itself is undefined, such as ln(x) at x = −1. Piecewise functions, summation notation, integrals inside the function and derivatives with respect to an expression rather than a variable are not supported. Very high derivatives of complicated inputs can grow too long to display, in which case the calculator asks you to lower the order. Numeric values use standard double precision: values at a point are rounded to 8 significant digits, and rounding noise such as 4 × 10⁻¹⁶ in place of 0 is shown as 0.

Tips for checking a derivative by hand

• Check the units: a derivative is output units per input unit, such as feet per second or dollars per item.
• Check the sign: if the graph rises at a point, the derivative there must be positive.
• Compare with the difference quotient: [f(a + 0.001) − f(a)]/0.001 should match f′(a) to about three digits for a smooth function.
• Differentiate the inside last: in chain-rule problems, write the outer derivative first and multiply by the inner derivative at the end, the same order the step list follows.
• Simplify before you differentiate when it helps: x³/x is x², whose derivative is 2x, with no quotient rule required.

Related calculators

• To find antiderivatives and areas under a curve, the reverse of differentiation, use the Integral Calculator.
• For limits, including the one inside the definition of the derivative, use the Limit Calculator.
• To plot a function and see where its slope is zero, use the Graphing Calculator.
• For polynomials built from repeated derivatives at one point, use the Taylor Series Calculator.
• For the slope between two points, the secant version of the derivative, use the Slope Calculator.
• To turn a slope and a point into a line equation, use the Point Slope Form Calculator.
• For the turning point of a parabola, where the derivative is zero, use the Vertex Calculator.

Sources

Derivatives are exact mathematical results, so no statistics are involved. The definitions and rules used on this page match the NIST Digital Library of Mathematical Functions, the U.S. government's reference for mathematical formulas:

• NIST DLMF §1.4, Calculus of One Variable: the derivative as a limit (1.4.4), the product and quotient rules (1.4.6, 1.4.7) and the chain rule (1.4.10).
• NIST DLMF §1.5, Calculus of Two or More Variables: partial derivatives (1.5.3), equal mixed partials (1.5.6) and the implicit function theorem.
• NIST DLMF §4.7, Derivatives of logarithms, exponentials and powers.
• NIST DLMF §4.20, Derivatives of the trigonometric functions.
• NIST DLMF §4.24, Inverse trigonometric functions: derivatives in equations 4.24.7 to 4.24.9.

Every number in the worked examples and tables above was computed with the same engine the calculator uses.