What this tool does
The Derivative Calculator computes the derivative of any mathematical expression and walks you through every step of the solution. Enter a function like x^3 + 2x^2 - 5x + 3 or sin(x)*cos(x), choose your variable and the order of differentiation, and the AI-powered engine returns the full derivative with a step-by-step breakdown of which rules were applied and why.
Whether you need a first, second, or third derivative, this tool handles polynomials, trigonometric functions, exponential and logarithmic expressions, and combinations of all of them. It shows domain restrictions when the result is not defined for all real numbers.
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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