# Derivative Calculator

> Compute derivatives with step-by-step solutions using power, chain, product, and quotient rules

**Category:** Math
**Keywords:** derivative, calculus, differentiation, power rule, chain rule, product rule, quotient rule, step by step, math solver, calculus solver
**URL:** https://complete.tools/derivative-calculator

## How derivatives work

A derivative measures the instantaneous rate of change of a function at any given point. Geometrically, it gives the slope of the tangent line to the curve at that point. The notation f'(x) or dy/dx both refer to the derivative of y with respect to x.

**Power Rule**
The most fundamental rule: if f(x) = x^n, then f'(x) = n * x^(n-1). For example, the derivative of x^4 is 4x^3. This rule applies to any term where a variable is raised to a constant power.

**Sum and Constant Rules**
Derivatives distribute across addition and subtraction. Constants differentiate to zero. So the derivative of 3x^2 + 7x - 4 is computed term by term: 6x + 7.

**Product Rule**
When two functions are multiplied, use (f*g)' = f'*g + f*g'. For example, the derivative of x^2 * sin(x) is 2x * sin(x) + x^2 * cos(x).

**Quotient Rule**
For a ratio of functions, (f/g)' = (f'*g - f*g') / g^2. For example, the derivative of sin(x)/x is (cos(x)*x - sin(x)) / x^2.

**Chain Rule**
Used when one function is nested inside another. If h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x). For example, the derivative of sin(x^2) is cos(x^2) * 2x.

## Common derivative formulas

These standard results are building blocks for more complex derivatives:

- d/dx [c] = 0 (constant)
- d/dx [x^n] = n * x^(n-1) (power rule)
- d/dx [e^x] = e^x (exponential)
- d/dx [a^x] = a^x * ln(a) (general exponential)
- d/dx [ln(x)] = 1/x (natural log)
- d/dx [sin(x)] = cos(x)
- d/dx [cos(x)] = -sin(x)
- d/dx [tan(x)] = sec^2(x)
- d/dx [arcsin(x)] = 1 / sqrt(1 - x^2)
- d/dx [arctan(x)] = 1 / (1 + x^2)

Higher-order derivatives are computed by applying differentiation repeatedly. The second derivative f''(x) describes concavity; the third derivative relates to the rate of change of acceleration in physics.

## How to use

1. Type your expression into the input field using standard notation (e.g., x^3 + 2*x^2, sin(x)*cos(x), e^(x^2), ln(x^2 + 1)).
2. Select the variable to differentiate with respect to (default is x; t, y, u, z also available).
3. Choose the order: 1st, 2nd, or 3rd derivative.
4. Click "Compute Derivative" and wait for the AI to process your expression.
5. Review the hero result card showing the final derivative.
6. Scroll through the step-by-step timeline to see every rule applied in order.
7. Check the domain restriction alert if your result has restrictions (e.g., undefined at x = 0).

## FAQs

**Q:** What notation should I use for entering expressions?


**A:** Use standard programming-style math notation. Write x^2 for x squared, x^(1/2) for square root, sin(x), cos(x), tan(x) for trig, e^x for the exponential function, and ln(x) for the natural log. Multiplication can be written as x*y or implied by adjacency in simple cases like 2x.


**Q:** Can this handle implicit differentiation or partial derivatives?


**A:** This tool computes explicit derivatives with respect to a single chosen variable. For partial derivatives, select the variable you are differentiating with respect to and treat all other variables as constants.


**Q:** Why does my result show a domain restriction?


**A:** Some derivatives are undefined at certain values. For example, the derivative of ln(x) is 1/x, which is undefined at x = 0. The tool notes these restrictions so you can interpret your result correctly.


**Q:** What is the difference between the first and second derivative?


**A:** The first derivative f'(x) gives the slope or instantaneous rate of change. The second derivative f''(x) tells you whether the function is concave up (f'' > 0) or concave down (f'' < 0) at each point, and is used to identify inflection points and optimize functions.


**Q:** Why do the intermediate steps look different from my textbook?


**A:** The AI may group or reorder terms differently while still producing a mathematically equivalent result. If you need a specific form, the "simplified" result card shows the most reduced version.

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