d/dx — Step-by-Step Differentiation
Last reviewed: April 2026
A derivative calculator computes the derivative of a mathematical function step by step, applying rules like the power rule, chain rule, product rule, and quotient rule. It is an essential tool for calculus students learning differentiation techniques.
The derivative of a function measures its instantaneous rate of change at any point. Geometrically, it gives the slope of the tangent line to the curve. If f(x) represents position, then f'(x) represents velocity. This calculator applies standard differentiation rules — power rule, sum rule, trig derivatives, exponential and logarithmic rules, and the chain rule — showing each step so you can follow the process. To visualize derivatives as slopes, graph both f(x) and f'(x) in our Graphing Calculator.
The most fundamental rule: d/dx[xⁿ] = n·xⁿ⁻¹. Bring the exponent down as a coefficient, then subtract 1 from the exponent. This works for any real exponent, including fractions (which is how you differentiate √x = x^(1/2)) and negatives (which handles 1/x = x^(-1)). Combined with the constant multiple rule, it handles any polynomial. For example, d/dx[3x⁴] = 3·4·x³ = 12x³. For solving polynomial equations, use our Equation Solver.
The six basic trig derivatives are: d/dx[sin x] = cos x, d/dx[cos x] = −sin x, d/dx[tan x] = sec²x, d/dx[cot x] = −csc²x, d/dx[sec x] = sec x·tan x, d/dx[csc x] = −csc x·cot x. When the argument is something other than plain x (like sin(3x) or cos(x²)), the chain rule applies: multiply by the derivative of the inner function. Our Scientific Calculator can evaluate trig functions at specific values.
The chain rule handles composite functions: d/dx[f(g(x))] = f'(g(x)) · g'(x). In words: differentiate the outer function (keeping the inner function unchanged), then multiply by the derivative of the inner function. For example, d/dx[sin(3x)] = cos(3x) · 3. The chain rule is by far the most-used rule in applied calculus because most real-world functions are compositions. For more calculus-related tools, see our Math Calculator collection.
| Function f(x) | Derivative f'(x) | Rule Name |
|---|---|---|
| x^n | n·x^(n-1) | Power rule |
| e^x | e^x | Exponential |
| ln(x) | 1/x | Natural log |
| sin(x) | cos(x) | Trig |
| cos(x) | -sin(x) | Trig |
| f(g(x)) | f'(g(x))·g'(x) | Chain rule |
The chain rule is the most important advanced differentiation technique, handling functions nested inside other functions. If y = f(g(x)), then dy/dx = f'(g(x)) times g'(x). In words: differentiate the outer function, leave the inner function unchanged, then multiply by the derivative of the inner function. For example, to differentiate sin(3x^2), identify the outer function as sine and the inner function as 3x^2. The derivative of sin(u) is cos(u), so f'(g(x)) = cos(3x^2). The derivative of 3x^2 is 6x. The chain rule gives cos(3x^2) times 6x = 6x cos(3x^2). The chain rule extends to multiple layers of nesting by applying the same process repeatedly from the outside inward.
When two functions are multiplied together, the derivative requires the product rule: d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x). This is often stated as "derivative of the first times the second, plus the first times the derivative of the second." For example, d/dx[x^2 sin(x)] = 2x sin(x) + x^2 cos(x). The quotient rule handles divisions: d/dx[f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2, often remembered as "low d-high minus high d-low, over low squared." Many students find quotient rule problems error-prone due to the subtraction in the numerator, so an alternative approach is to rewrite f/g as f times g^(-1) and apply the product rule with the chain rule instead.
| Function f(x) | Derivative f'(x) | Rule Applied |
|---|---|---|
| x^n | n x^(n-1) | Power rule |
| e^x | e^x | Exponential |
| ln(x) | 1/x | Logarithmic |
| sin(x) | cos(x) | Trig |
| cos(x) | -sin(x) | Trig |
| tan(x) | sec^2(x) | Trig |
| a^x | a^x ln(a) | Exponential |
| arcsin(x) | 1/sqrt(1-x^2) | Inverse trig |
| arctan(x) | 1/(1+x^2) | Inverse trig |
Derivatives appear throughout the physical sciences under different names. In physics, the derivative of position with respect to time is velocity, and the derivative of velocity is acceleration. In chemistry, reaction rates are derivatives of concentration with respect to time. In economics, marginal cost, marginal revenue, and marginal utility are all derivatives that describe how one quantity changes in response to small changes in another. In engineering, control systems use derivatives to anticipate and correct errors before they grow large, which is the "D" in PID controllers found in everything from cruise control to industrial temperature regulation.
Optimization problems are among the most practical applications of derivatives. Finding the maximum or minimum of a function requires setting its derivative equal to zero and solving for the critical points. The second derivative test then determines whether each critical point is a maximum, minimum, or neither. Real-world optimization examples include maximizing the volume of a box given a fixed amount of material, minimizing the cost of a pipeline that crosses a river at an angle, and finding the production level that maximizes profit. These problems combine differentiation with algebraic modeling and appear in every quantitative field. For integration, the inverse operation of differentiation, see our Integral Calculator, and for limit evaluation, our Limit Calculator.
Not every function can be written in the form y = f(x). Equations like x^2 + y^2 = 25 (a circle) define y implicitly as a function of x. Implicit differentiation differentiates both sides with respect to x, treating y as a function of x and applying the chain rule whenever y appears. For the circle equation, differentiating gives 2x + 2y(dy/dx) = 0, so dy/dx = -x/y. This technique is essential for finding tangent lines to curves defined by equations that cannot be solved explicitly for y.
Related rates problems extend implicit differentiation to situations where multiple quantities change simultaneously over time. If a ladder slides down a wall, the rates of change of the top height and bottom distance are related through the Pythagorean theorem. If a balloon is being inflated, the rates of change of radius and volume are connected through the volume formula. The key steps are: identify the relationship between variables, differentiate implicitly with respect to time, substitute known values, and solve for the unknown rate. These problems model real physical situations and appear frequently in physics and engineering. Explore related mathematical tools with our Equation Solver and Logarithm Calculator.
Taking the derivative of a derivative produces the second derivative, written f''(x). The second derivative measures the rate of change of the rate of change, which corresponds to concavity. When f''(x) is positive, the function is concave up (bowl-shaped); when negative, concave down (dome-shaped). Points where f''(x) changes sign are inflection points where the concavity reverses. Higher-order derivatives continue this pattern: the third derivative of position with respect to time is called jerk in physics because it describes how abruptly acceleration changes, which passengers feel as jerky motion in vehicles.
Taylor series use derivatives of all orders to approximate any smooth function as a polynomial. This powerful technique allows complex functions like sine, cosine, and exponential to be computed using only addition and multiplication. The Maclaurin series (Taylor series centered at zero) for the exponential function is 1 + x + x-squared/2! + x-cubed/3! and so on, converging for all real numbers. Truncating the series after a few terms provides a polynomial approximation whose accuracy improves with more terms and degrades as x moves away from the center point. Scientific calculators use these polynomial approximations internally to compute trigonometric and exponential functions. For computing definite integrals, see our Integral Calculator.
Functions of multiple variables require partial derivatives, where you differentiate with respect to one variable while treating all others as constants. For a function of x and y, the partial derivative with respect to x treats y as a constant, and vice versa. Partial derivatives are used extensively in physics (electric fields, fluid dynamics, heat transfer), economics (marginal analysis with multiple inputs), machine learning (gradient descent optimizes loss functions with thousands of parameters), and engineering (stress and strain analysis in materials).
The gradient vector, which collects all partial derivatives, points in the direction of steepest increase of a function, making it the foundation of optimization algorithms used in artificial intelligence. Neural networks with millions of parameters are trained by computing partial derivatives of the loss function with respect to each weight using backpropagation, an efficient application of the chain rule, then adjusting weights in the direction that reduces loss. This process repeats millions of times during training, making derivatives arguably the single most important mathematical operation in modern AI. For matrix-related calculations, explore our Matrix Calculator.
See also: Graphing Calculator · Equation Solver · Scientific Calculator · Quadratic Solver · Logarithm Calculator · Slope Calculator
→ Know the basic rules by heart. Power rule (d/dx xⁿ = nxⁿ⁻¹), product rule, quotient rule, and chain rule cover 95% of derivatives you'll encounter. The calculator shows which rule applies at each step.
→ Use the second derivative for concavity. f''(x) > 0 means the function is concave up (like a cup), f''(x) < 0 means concave down. Inflection points occur where f''(x) changes sign.
→ Check your answer by graphing. The derivative should be zero at peaks and valleys of the original function, and positive where the function is increasing. Verify visually with our Graphing Calculator.
→ Apply to optimization problems. Set f'(x) = 0 to find critical points, then use the second derivative test to determine if each is a maximum or minimum.
See also: Integral Calculator · Graphing Calculator · Limit Calculator · Scientific Calculator
Derivatives are the mathematical foundation of optimization — finding the best possible outcome given constraints. In economics, marginal cost and marginal revenue are derivatives of total cost and total revenue functions; setting marginal revenue equal to marginal cost determines the profit-maximizing quantity. In engineering, derivatives model rates of change in electrical circuits (current as the derivative of charge), mechanical systems (velocity as the derivative of position), and thermodynamics (heat transfer rates as derivatives of temperature distributions). Machine learning relies heavily on derivatives through gradient descent algorithms that iteratively minimize loss functions by following the negative gradient toward optimal parameter values.
Related rates problems demonstrate how derivatives connect changing quantities in the physical world. If a spherical balloon is inflated at 100 cubic centimeters per second, how fast is its radius increasing when the radius is 10 cm? The chain rule connects the volume rate of change (known) to the radius rate of change (unknown) through the volume formula V = (4/3)πr³. Medical dosing models use derivatives to describe how drug concentration changes in the bloodstream over time — the rate of elimination determines dosing frequency. Derivatives of production functions reveal diminishing returns, helping manufacturers decide when adding more workers or capital stops being cost-effective. This calculator handles the symbolic differentiation, but understanding what the derivative means in context is what transforms a mathematical result into actionable insight.