Derivative Calculator

A derivative calculator is a tool that applies calculus differentiation rules automatically to find the exact derivative of any mathematical function.

Instead of manually applying the Power Rule, Chain Rule, Product Rule, or Quotient Rule, you simply enter your function and get the answer instantly with full step-by-step working.

Our calculator applies these rules automatically and shows you exactly which rule was used at each step:

• The Power Rule (d/dx xⁿ = nxⁿ⁻¹) handles polynomials.
• The Chain Rule handles composite functions.
• The Product and Quotient Rules handle multiplication and division of functions.

A derivative measures the instantaneous rate of change of a function — the slope of the line tangent to its graph at a single point.

Formally, the derivative f'(x) is the limit of the difference quotient [f(x+h) − f(x)] / h as h approaches 0. When that limit exists, the function is differentiable at x. As Wolfram MathWorld notes, the derivative is the foundational object of differential calculus.

Geometrically, if you zoom infinitely close to a smooth curve it looks like a straight line, and the derivative gives that line's slope. Physically, if f(x) is position, then f'(x) is velocity — how fast position changes at each instant.

Four rules cover most differentiation:

• The Power Rule: d/dx(xⁿ) = n·xⁿ⁻¹, so d/dx(x⁴) = 4x³.
• The Product Rule: d/dx(u·v) = u'v + uv'.
• The Quotient Rule: d/dx(u/v) = (u'v − uv')/v².
• The Chain Rule for composites: d/dx[f(g(x))] = f'(g(x))·g'(x).

Khan Academy's differentiation course walks through each with worked examples. For instance, d/dx(x²·sin x) = 2x·sin x + x²·cos x by the Product Rule. These rules combine freely with the Sum Rule, which lets you differentiate a polynomial term by term.

To differentiate f(x) = x³ + 2x² − 5x + 3, apply the Sum Rule and handle each term with the Power Rule: d/dx(x³) = 3x², d/dx(2x²) = 4x, d/dx(−5x) = −5, and d/dx(3) = 0 because constants have zero rate of change. Adding these gives f'(x) = 3x² + 4x − 5.

For composite functions like sin(2x), use the Chain Rule: the outer derivative cos(2x) times the inner derivative 2 gives 2·cos(2x). This calculator reproduces exactly this decomposition, labeling each rule so you can check your own hand computation against a verified result.

Beyond polynomials, standard functions have memorized derivatives.

For trigonometry: d/dx(sin x) = cos x, d/dx(cos x) = −sin x, and d/dx(tan x) = sec²x, as tabulated in the NIST Digital Library of Mathematical Functions (DLMF). The exponential function is uniquely self-derivative: d/dx(eˣ) = eˣ. The natural logarithm gives d/dx(ln x) = 1/x for x > 0.

Combined with the Chain Rule, these extend easily: d/dx(e^(2x)) = 2e^(2x) and d/dx(ln(x²)) = 2/x. Encyclopaedia Britannica describes these elementary derivatives as the building blocks that, together with the four rules, differentiate nearly any expression you meet in calculus.

The second derivative f''(x) is simply the derivative of the first derivative, and it measures how the rate of change itself changes. If f(x) is position, f'(x) is velocity and f''(x) is acceleration. For f(x) = x³, the first derivative is 3x², and differentiating again gives f''(x) = 6x.

The second derivative also reveals concavity: f''(x) > 0 means the curve bends upward (concave up), while f''(x) < 0 means concave down, and a sign change marks an inflection point.

Continuing this process yields third, fourth, and higher-order derivatives, written f'''(x) or f⁽ⁿ⁾(x). This tool computes higher-order derivatives by repeated symbolic differentiation.

Derivatives quantify change wherever it occurs:

• In physics, velocity is the derivative of position and acceleration is the derivative of velocity — the framework Newton and Leibniz built, as Encyclopaedia Britannica recounts.
• In economics, marginal cost and marginal revenue are derivatives of total cost and total revenue functions.
• In optimization, setting f'(x) = 0 locates maxima and minima, which powers everything from engineering design to machine-learning gradient descent.
• In biology, continuous growth rates of populations are modeled with derivatives, whereas the classic discrete rabbit-breeding sequence behind our fibonacci calculator captures growth generation by generation.
• Even curve tracing in computer graphics relies on tangent slopes.

Understanding differentiation therefore unlocks a huge range of applied problems across the sciences, finance, and technology.

Because a derivative gives the slope of the tangent line, you can find where a function levels off by solving f'(x) = 0; these are the critical points.

Khan Academy's calculus curriculum teaches the first-derivative test: if f' changes from positive to negative at a critical point, that point is a local maximum; from negative to positive gives a local minimum. The second derivative offers a shortcut — the second-derivative test — where f''(x) < 0 confirms a maximum and f''(x) > 0 confirms a minimum.

To find the slope of a curve at a specific point, simply evaluate f'(x) at that x-value, which this calculator does automatically when you supply an evaluation point.

Several errors trip up students most often:

• The most frequent error is forgetting the Chain Rule's inner derivative: d/dx(sin(2x)) is 2·cos(2x), not cos(2x).
• Another is misapplying the Power Rule to exponentials — d/dx(eˣ) = eˣ, not x·eˣ⁻¹.
• Many students confuse the Product Rule with simply multiplying derivatives; d/dx(u·v) is u'v + uv', not u'·v'.
• Sign errors are common with d/dx(cos x) = −sin x.
• Constants inside functions are sometimes dropped, and dividing before differentiating a quotient can invert the Quotient Rule's numerator order.

Wolfram MathWorld's worked examples are a good cross-check. Whenever a hand result disagrees with this calculator's step-by-step output, one of these slips is usually the cause.

This tool performs symbolic differentiation, meaning it manipulates the algebraic expression itself to return an exact formula such as 3x² + 4x − 5, rather than approximating a slope numerically.

Numerical differentiation instead estimates f'(x) with finite differences like [f(x+h) − f(x)] / h for a small h, which introduces rounding and truncation error. Symbolic results are exact and reusable — you can differentiate again or evaluate at any point without accumulating error.

Symbolic computation is the approach used by computer-algebra systems and described in the NIST DLMF for closed-form results. For exact homework answers, tangent-line slopes, and higher-order derivatives, symbolic differentiation is the reliable choice this calculator provides.