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📘 About the Derivative Calculator with Steps
The derivative measures how fast a function changes: the slope of its graph at each point. It is the first tool of calculus, used for rates of change, velocity, optimisation (finding maxima and minima) and curve sketching.
This calculator differentiates symbolically, the way you would on paper, using the power, product, quotient and chain rules and the derivatives of sin, cos, tan, eˣ, ln x and inverse trig functions. It simplifies the answer, shows the rule used for each term, finds second and higher derivatives, and gives the slope and tangent line at a point.
🧮 Formula
✏️ Worked examples
Product rule: x³ sin x
You get| Derivative | f′(x) = x³·cos(x) + 3x²·sin(x) |
| Function | f(x) = x³·sin(x) |
f′(x) = x³·cos(x) + 3x²·sin(x).
Chain rule at a point
You get| Derivative | f′(x) = 2x/(x² + 1) |
| Function | f(x) = ln(x² + 1) |
| Value at x = 1 | f′(1) = 1 |
| Tangent line at (1, 0.6931471806) | y = x − 0.3068528194 |
| Normal line | y = 1.693147181 − x |
f′(x) = 2x/(x² + 1); slope at x = 1: y = x − 0.3068528194.
🧭 How to use this calculator
What you enter
- Function f(x)Use ^ for powers, * or nothing for ×: 3x^2 + 2sin(x), e^(2x), sqrt(x+1), ln(x).
- VariableOne letter. Other letters are treated as constants.
- Which derivative
- Evaluate at x = (optional)Gives the slope there and the tangent line.
❓ Frequently asked questions
How do you find the derivative of a function?
Split it into terms and apply the rules: power rule for xⁿ, the constant multiple rule, product and quotient rules for products and fractions, and the chain rule for a function inside another (like sin(3x) or (x² + 1)⁵).
What is the derivative of eˣ, ln x, sin x and cos x?
d/dx eˣ = eˣ, d/dx ln x = 1/x, d/dx sin x = cos x and d/dx cos x = −sin x. The derivative of tan x is sec²x.
What does the second derivative tell you?
Concavity: f″ > 0 means the graph curves upward (a minimum when f′ = 0), f″ < 0 means it curves downward (a maximum). In physics, the second derivative of position is acceleration.
How do I find the equation of the tangent line?
Find the slope m = f′(a), then y − f(a) = m(x − a). Enter a point in the calculator to get it directly.
