differentiate function calculator

What this differentiate function calculator does

This tool computes the derivative of a mathematical expression with respect to a chosen variable. In plain terms, it tells you how quickly a function is changing at each input value. You can also evaluate that derivative at a specific point to get an exact slope value.

• Find symbolic derivatives for many common functions
• Show both raw and simplified derivative forms
• Compute slope at a point (for tangent line applications)
• Handle algebraic, trigonometric, exponential, and logarithmic expressions

How to use the calculator

1) Enter your function

Use standard math syntax like x^3, sin(x), e^x, or (x^2+1)/(x-4). Write explicit multiplication with * (for example, 3*x instead of 3x).

2) Choose the variable

Most users leave this as x. If your function uses another variable, like t or y, enter it there.

3) Optional: evaluate at a point

If you provide a value such as 2 or pi/4, the calculator will return the numeric slope of the curve at that input.

Supported syntax and function names

The calculator accepts operators and function names commonly used in calculus classes and engineering tools.

• Operators: +, -, *, /, ^
• Trig: sin, cos, tan, sec, csc, cot
• Inverse trig: asin, acos, atan
• Exponential/log: exp, e^x, log (natural log), ln is also accepted
• Roots/other: sqrt(x), abs(x)
• Constants: pi, e

Core differentiation rules used

Under the hood, symbolic differentiation uses standard calculus identities:

• Power rule: d/dx[x^n] = n*x^(n-1)
• Constant multiple rule
• Sum and difference rules
• Product rule
• Quotient rule
• Chain rule for nested functions like sin(x^2)

Example inputs to try

• x^5 - 4*x + 1
• sin(x) + cos(x)
• x*log(x)
• (x^2 + 1)/(x - 3)
• exp(2*x)
• sqrt(x^2 + 9)

Common mistakes and quick fixes

Missing multiplication symbols

Write 2*x, not 2x. Also write x*sin(x), not xsin(x).

Wrong logarithm format

Use log(x) for natural logarithm. If you type ln(x), this calculator automatically converts it.

Invalid point values

If evaluating at a point, make sure the function is defined there. For example, avoid dividing by zero or taking log of a non-positive value.

Why derivatives matter

Derivatives are the language of change. They appear in velocity and acceleration calculations, optimization problems in business, curve fitting in data science, and sensitivity analysis in engineering and economics. A reliable differentiate function calculator helps you move faster while still checking your understanding of the underlying rules.