calculator of derivatives

Derivative Calculator

Enter a function, choose the variable, and optionally evaluate the slope at a specific point.

Function f(x)

Use math syntax like x^2, sin(x), ln(x), exp(x), sqrt(x).

Differentiate with respect to

Evaluate at point (optional)

If provided, the calculator returns both f(value) and f'(value).

Try examples:

What a Derivative Tells You

A derivative measures how quickly one quantity changes with respect to another. In plain language, it is the slope of a curve at a single point. If you are looking at position over time, the derivative gives velocity. If you are looking at cost over production volume, the derivative gives marginal cost.

That is why derivatives appear everywhere: physics, economics, optimization, machine learning, business forecasting, and engineering design. They turn a static formula into a dynamic story about change.

How to Use This Calculator

1) Enter your function

Type a valid expression such as x^3 + 2*x, sin(x), or ln(x). Use the multiplication symbol * when needed.

2) Choose the variable

Most users differentiate with respect to x, but you can use t, y, or another symbol if your expression uses it.

3) Optionally evaluate at a point

Enter a numeric value to find the instantaneous slope there. This is useful for tangent line problems and quick checks in optimization tasks.

Common Derivative Rules (Quick Reference)

Power Rule: d/dx[xⁿ] = n·x^n-1
Constant Rule: d/dx[c] = 0
Sum Rule: d/dx[f + g] = f' + g'
Product Rule: d/dx[fg] = 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)

Useful Function Syntax

Trigonometric: sin(x), cos(x), tan(x)
Inverse trig: asin(x), acos(x), atan(x)
Logarithms: ln(x), log(x, 10)
Exponential: exp(x), 2^x
Roots: sqrt(x)

Worked Examples

Example A: Polynomial

For f(x) = x^3 + 2x - 5, the derivative is f'(x) = 3x^2 + 2. At x = 2, slope is f'(2) = 14.

Example B: Trigonometric Product

For f(x) = sin(x)cos(x), apply the product rule: f'(x) = cos(x)cos(x) - sin(x)sin(x), which also simplifies to cos(2x).

Example C: Exponential over Quadratic

For f(x) = exp(2x)/(1 + x^2), use quotient + chain rules. This is exactly the type of expression where a calculator saves time and reduces algebra mistakes.

Practical Applications

Business: Optimize revenue, cost, and profit.
Data science: Understand gradient-based learning methods.
Physics: Convert position to velocity and acceleration.
Finance: Approximate sensitivity and local rates of change.
Engineering: Analyze system response and control behavior.

Common Mistakes to Avoid

Forgetting parentheses, e.g., writing sin x^2 instead of sin(x^2).
Mixing variable names between the expression and derivative variable field.
Confusing ^ (power) with * (multiplication).
Evaluating at points outside a function’s domain (such as ln(-1)).

Final Thoughts

Derivatives are a foundational tool for understanding change. Once you can compute and interpret them quickly, many advanced topics become easier: optimization, differential equations, and machine learning among them.

Use the calculator above to experiment with your own functions and build intuition through practice.