What a Derivative Means
A derivative describes how a function changes as its input changes. Geometrically, it is the slope of the tangent line at a point on a curve. In applied settings, derivatives represent rates: speed as the derivative of position, marginal cost as the derivative of total cost, growth rates in biology, and sensitivity measures in engineering and finance.
This Derivative Calculator computes symbolic derivatives using standard calculus rules and also produces numeric evaluations and derivative tables so you can analyze change over a range of inputs.
Symbolic Differentiation and Why It Matters
Symbolic differentiation produces an exact algebraic formula for the derivative. Unlike numeric approximations, symbolic results can be simplified, evaluated precisely (where defined), and differentiated again for higher-order analysis such as concavity and curvature.
For example, if f(x)=x², symbolic differentiation yields f′(x)=2x. From that single expression, you can immediately compute slope at any x, find turning points, and study behavior without recalculating from scratch.
Common Differentiation Rules
Most derivatives can be computed by combining a small set of rules:
- Constant rule: d/dx(c)=0
- Power rule: d/dx(xⁿ)=n·xⁿ⁻¹
- Sum rule: d/dx(f+g)=f′+g′
- Product rule: d/dx(f·g)=f′g+fg′
- Quotient rule: d/dx(f/g)=(f′g−fg′)/g²
- Chain rule: d/dx(f(g(x)))=f′(g(x))·g′(x)
The calculator applies these rules to build a derivative expression. The Steps table is designed to be readable and instructional, showing which rule drives each transformation.
Higher-Order Derivatives
Higher-order derivatives measure higher-level behavior. The second derivative f″(x) describes concavity (curvature) and is used to identify minima and maxima. The third derivative can describe how curvature changes, and higher orders appear in Taylor series, physics, and numerical methods.
In this tool, you can select derivative order and compute f′, f″, f‴, and beyond by repeated differentiation.
Partial Derivatives for Multivariable Functions
If your function includes multiple variables, a partial derivative differentiates with respect to one variable while treating the others as constants. This is central to multivariable calculus, optimization, gradients, and machine learning (where loss functions depend on many parameters).
Choose the variable in the dropdown (x, y, t, u, v) and write your expression using those variables. The calculator will differentiate with respect to the selected one.
Evaluating f(x) and f′(x) at a Point
A symbolic derivative becomes immediately useful when evaluated at a specific x. For example, if f′(2)=10, the function is increasing at x=2 with a slope of 10, meaning small increases in x produce large increases in f(x) around that point.
The Evaluate tab computes f(x) and f′(x) numerically at your chosen x value (when the expression is defined). This helps confirm results and connect formulas to actual values.
Derivative Tables for Analysis and Plotting
A derivative table lists x values with both f(x) and f′(x). This is useful for:
- Plotting a function and its slope in a spreadsheet or graphing tool
- Finding where the function changes fastest
- Checking sign changes in the derivative (increasing vs decreasing)
- Exploring curvature by using higher-order derivatives (via the order selector)
The table can be exported to CSV for easy use in Excel, Google Sheets, or statistical tools.
Limitations and Input Tips
Symbolic calculus engines can be complex. This calculator focuses on common classroom and practical expressions with readable output: polynomials, roots, exponentials, logs, and standard trigonometric functions. For best results:
- Use explicit multiplication: 2*x instead of 2x
- Use parentheses: sin(x^2) instead of sin x^2
- Use ln(x) for natural log and exp(x) for e^x
- Keep expressions well-formed and avoid ambiguous notation
If an expression is undefined at a value (like ln(x) at x≤0), evaluation results will reflect that.
FAQ
Derivative Calculator – Frequently Asked Questions
Answers about derivatives, rules, steps, higher-order derivatives, and partial derivatives.
A derivative measures how a function changes as its input changes. It represents the slope of the function at a point and is used to analyze rates of change, slopes, optimization, and motion.
Symbolic differentiation finds an exact algebraic expression for the derivative, using calculus rules (like the power rule and chain rule), instead of approximating slopes numerically.
This calculator uses standard derivative rules including constant and power rules, sum rule, product rule, quotient rule, and chain rule. It also supports common functions like polynomials, exponentials, logarithms, and trig functions.
Yes. Choose the derivative order (first derivative, second derivative, etc.). The calculator applies differentiation repeatedly to compute higher-order derivatives.
Yes. It performs practical simplifications (like combining constants and removing unnecessary terms) so results are easier to read.
Yes. Enter a numeric x value and the calculator will compute f(x) and f′(x) at that point (when the expression is defined).
Yes. Enter a multivariable function and choose the variable to differentiate with respect to. The other variables are treated as constants.
Use standard math syntax: x^2, sin(x), cos(x), tan(x), ln(x), log(x), e^x as exp(x), sqrt(x). Use * for multiplication when needed (e.g., 2*x).
Yes. Build a table of x values with f(x) and f′(x) and export it to CSV for plotting or analysis in a spreadsheet.