Symbolic differentiation with step-by-step rules and interactive graph
Calculus is one of the most powerful branches of mathematics, and at its heart lies the concept of the derivative. A derivative tells you the instantaneous rate of change of a function — how fast a quantity is growing, shrinking, or changing at any given moment. Whether you are a student working through your first calculus course, an engineer analyzing how a system responds to small perturbations, a physicist modeling motion, or a data scientist studying the gradient of a loss function, understanding derivatives is fundamental. Our Derivative Calculator makes this process effortless. Enter any mathematical expression, choose your differentiation variable, select the order of the derivative, and the calculator instantly returns the exact symbolic result along with a detailed step-by-step breakdown of every rule applied. You can also evaluate the derivative at a specific point to find the exact slope of a tangent line, visualize both the original function and its derivative on an interactive graph, and review the relevant differentiation rules — all without leaving the page. Differentiation follows a set of well-defined rules that mathematicians developed over centuries. The Power Rule handles polynomial terms: the derivative of xⁿ is n·xⁿ⁻¹. The Product Rule handles expressions that are products of two functions. The Quotient Rule handles rational expressions — fractions of two functions. The Chain Rule, arguably the most important, handles composite functions: if you have f(g(x)), the derivative is f′(g(x))·g′(x). Beyond these fundamental rules, there are specific derivatives for all standard transcendental functions: sin(x), cos(x), tan(x), arcsin(x), arccos(x), arctan(x), eˣ, aˣ, ln(x), log₁₀(x), and all hyperbolic functions sinh, cosh, tanh. Higher-order derivatives are also essential: the second derivative f″(x) describes acceleration (concavity of the graph), the third and beyond appear in Taylor series, physics, and engineering analysis. Our calculator supports up to the 10th-order derivative through repeated symbolic differentiation, making it suitable for advanced coursework and professional use. For students, seeing the step-by-step derivation is invaluable. Rather than just getting an answer, you see every rule applied in sequence, color-coded by which rule was used — Power Rule in indigo, Chain Rule in violet, Product Rule in amber. This pedagogical approach helps you learn the patterns of differentiation, not just get answers. For professionals and researchers, the calculator serves as a fast verification tool. When symbolic algebra becomes tedious, you can double-check your manual calculations instantly. The evaluation-at-a-point feature gives you the exact numerical slope at any x value, and the tangent line overlay on the graph makes it visually clear what the derivative means geometrically. The graph plots f(x) and f′(x) together over a customizable domain. You can see at a glance where the derivative is zero (potential extrema), where it is positive (function increasing), and where it is negative (function decreasing). If you specify an evaluation point, the tangent line to the curve is drawn at that point, illustrating the geometric interpretation of the derivative as the slope of the tangent. A built-in calculation history stores your last 10 computations, so you can revisit previous results without re-entering expressions. The quick-reference cheat sheet lists the most common derivatives for rapid lookup. One-click copy lets you transfer results into your notes, homework, or code instantly. This calculator is powered by math.js, a comprehensive JavaScript mathematics library that performs exact symbolic differentiation in the browser — no server required, no data sent anywhere, completely private and instant.
Understanding Derivatives
What Is a Derivative?
A derivative measures the instantaneous rate of change of a function with respect to one of its variables. Geometrically, f′(x) at a point x = a equals the slope of the tangent line to the graph of f at that point. Formally, f′(x) = lim(h→0) [f(x+h) − f(x)] / h. This limit definition, while precise, is laborious to compute directly for complex expressions. Instead, we use differentiation rules — systematic shortcuts derived from this limit that handle all standard function types. The derivative of a constant is 0. The derivative of x is 1. The derivative of x² is 2x. These simple cases generalize into powerful rules that handle any combination of operations.
How Are Derivatives Calculated?
Differentiation is applied via a hierarchy of rules. First, identify the top-level structure of the expression: is it a sum, product, quotient, or composition? Apply the corresponding rule. The Power Rule d/dx[xⁿ] = n·xⁿ⁻¹ covers polynomials. The Sum/Difference Rule d/dx[f ± g] = f′ ± g′ splits expressions into terms. The Product Rule d/dx[f·g] = f′·g + f·g′ handles multiplications. The Quotient Rule d/dx[f/g] = (f′·g − f·g′) / g² handles divisions. The Chain Rule d/dx[f(g(x))] = f′(g(x))·g′(x) handles compositions. Each transcendental function (sin, cos, exp, ln, etc.) has a known closed-form derivative. Higher-order derivatives are computed by applying differentiation repeatedly.
Why Derivatives Matter
Derivatives appear everywhere in science, engineering, economics, and computing. In physics, velocity is the derivative of position, and acceleration is the derivative of velocity. Engineers use derivatives to analyze system stability and optimize designs. Economists use marginal cost and marginal revenue — derivatives of cost and revenue functions. In machine learning, gradient descent minimizes a loss function by following the negative direction of its gradient (the derivative with respect to model parameters). In biology, population growth rates are derivatives. Derivatives are also essential for finding maxima and minima of functions — setting f′(x) = 0 and checking f″(x) identifies local extrema, which is the foundation of mathematical optimization.
Limitações e Escopo
This calculator uses math.js for symbolic differentiation and covers polynomials, standard trigonometric functions (sin, cos, tan, asin, acos, atan), hyperbolic functions (sinh, cosh, tanh), exponential (exp, e^x), and logarithmic functions (log, log10, log2). Complex expressions may occasionally produce unsimplified results or require reformatting. The calculator does not support implicit differentiation (treating y as a function of x) in this version, or special functions like erf, gamma, or Lambert W. For extremely large expression orders, simplification may produce longer forms. The step-by-step display identifies the primary rule at each level; nested composite functions may show simplified step descriptions. Numerical evaluation may return null for expressions with singularities at the chosen x value.
How to Use the Derivative Calculator
Enter Your Function
Type your mathematical function in the input field. Use ^ for exponents (x^2 means x²), * for multiplication (3*x means 3x), and standard function names like sin(x), cos(x), exp(x), log(x), and sqrt(x). Use the symbol toolbar buttons to insert common symbols instantly, or click one of the quick example buttons to load a preset function.
Choose Variable and Order
Select the variable to differentiate with respect to from the dropdown (x is the default, but y, z, t, u, v, and w are also available for multivariable expressions). Then choose the derivative order: 1st for f′(x), 2nd for f″(x), up to 10th for high-order analysis. The calculator computes higher-order derivatives by applying differentiation repeatedly.
Optionally Evaluate at a Point
To find the exact value of the derivative at a specific x value (for example, the slope of the tangent at x = 2), enter that number in the 'Evaluate at point' field. The calculator will show both f(a) and f′(a), and it will draw the tangent line at that point on the graph. Leave this field empty if you only need the symbolic derivative.
Read the Results and Steps
The main result card shows the simplified symbolic derivative. Below it, color-coded badges identify which differentiation rules were applied. The step-by-step panel walks through each rule application in sequence — perfect for learning. The graph shows f(x) and f′(x) plotted together over your specified range, making it easy to see where the function is increasing, decreasing, or has extrema.
Perguntas Frequentes
What is a derivative and what does it represent geometrically?
A derivative f′(x) measures how fast a function changes at any given point. Formally, it is the limit of the difference quotient [f(x+h) − f(x)] / h as h approaches zero. Geometrically, f′(a) equals the slope of the tangent line to the graph of f at the point (a, f(a)). If f′(a) > 0, the function is increasing at x = a. If f′(a) < 0, it is decreasing. If f′(a) = 0, the tangent is horizontal, which may indicate a local maximum, minimum, or saddle point. This geometric interpretation makes derivatives essential for optimization and curve analysis.
Which differentiation rules does this calculator apply?
The calculator applies all standard calculus differentiation rules automatically. These include: the Power Rule (d/dx[xⁿ] = n·xⁿ⁻¹), the Constant Rule (d/dx[c] = 0), the Constant Multiple Rule (d/dx[c·f] = c·f′), the Sum and Difference Rule (d/dx[f ± g] = f′ ± g′), the Product Rule (d/dx[f·g] = f′·g + f·g′), the Quotient Rule (d/dx[f/g] = (f′·g − f·g′) / g²), and the Chain Rule (d/dx[f(g(x))] = f′(g(x))·g′(x)). It also handles all standard transcendental functions — trigonometric, inverse trigonometric, exponential, and logarithmic — using their known closed-form derivatives.
How do I enter complex expressions like fractions and nested functions?
Use standard mathematical syntax accepted by the calculator. For fractions, write (numerator) / (denominator) — for example, (x^2 - 1) / (x + 1). For exponents, use ^ — for example, x^3 or e^(2*x). For nested functions like sin(x²), write sin(x^2). Always use * explicitly for multiplication — write 3*x, not 3x. Parentheses are important for grouping: sin(x^2) means sin applied to x², while sin(x)^2 means the square of sin(x). Use the symbol toolbar buttons to insert common symbols without typing them.
What does the evaluation at a point feature do?
When you enter a number in the 'Evaluate at point' field (for example, x = 2), the calculator computes two values: f(a), which is the value of the original function at that x value, and f′(a), which is the exact numerical slope of the derivative at that x value. This slope equals the steepness of the tangent line to the curve at x = a. The tangent line is also drawn on the graph as an orange line, giving you a visual illustration of what the derivative means at that specific point. This feature is especially useful for optimization problems and physics applications.
Can I compute second, third, and higher-order derivatives?
Yes. The derivative order selector lets you choose from 1st through 10th order. A second derivative f″(x) is the derivative of the first derivative and describes the concavity of the function — positive f″ means concave up (bowl shape), negative f″ means concave down (arch shape). The third derivative describes the rate of change of concavity (jerk in physics). Higher-order derivatives are computed by applying symbolic differentiation repeatedly, so each order starts from the result of the previous one. For simple polynomial functions, derivatives beyond the polynomial degree become zero.
What function syntax does the calculator support?
The calculator supports a wide range of function types. Polynomials: x^2, x^3, x^(1/2). Trigonometric: sin(x), cos(x), tan(x), sec(x), csc(x), cot(x). Inverse trig: asin(x), acos(x), atan(x). Hyperbolic: sinh(x), cosh(x), tanh(x). Exponential: exp(x) or e^x. Logarithmic: log(x) for natural log, log10(x) for base-10. Square root: sqrt(x). Constants: pi, e. Products, sums, and compositions of any of the above. For best results, always write multiplication explicitly with *, use parentheses for grouping, and use ^ for exponentiation. The symbol toolbar also provides one-click insertion of common operators.