Power Rule
\(\frac{d}{dx}[x^3] = 3x^2\)
Derivative Calculator
Calculate derivatives with step-by-step solutions and rules
Explore the derivative
Type a function in x or pick a preset. The blue curve is f(x), the green curve is f'(x), and the red tangent line is anchored at the slider's x-value.
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-5.005.00
Use x as the variable. Examples: x^3 - 3*x, sin(x), exp(x), 1/x.
Common functions
Predict what will happen
Where is f'(x) zero on x³ − 3x?
f'(x) = 3x² − 3. Set it to zero and solve.
Tangent at x₀
The red line through (x₀, f(x₀)) with slope f'(x₀) is the best linear approximation of f near x₀. It is the limit of secant lines as the second point slides into x₀.
Common mistake
Don't confuse the function f(x) (blue) with the derivative f'(x) (green). Where the blue curve is steep, the green curve is far from zero; where the blue curve is flat, the green curve crosses zero.
Why it works
f'(x₀) is defined as the limit of the difference quotient (f(x₀+h) − f(x₀)) / h as h → 0. The tangent line replaces the curve with that linear behavior locally.
Results
Final Answer
\(f'(x) = 3 * x ^ 2 - 3\)
Step-by-step Solution
- Function \(f(x) = x ^ 3 - 3 * x\)
- Combine results \(f'(x) = 3 * x ^ 2 - 3\)
- At x = 1, \(f(1) = -2\), \(f'(1) = 0\)
Theory & Formula
The derivative measures the rate of change of a function at any given point. It represents the slope of the tangent line to the function at that point.
Common derivative rules:
- Power Rule: \(\frac{d}{dx}[x^n] = nx^{n-1}\)
- Constant Rule: \(\frac{d}{dx}[c] = 0\)
- Sum Rule: \(\frac{d}{dx}[f + g] = f' + g'\)
- Product Rule: \(\frac{d}{dx}[fg] = f'g + fg'\)
- Trigonometric: \(\frac{d}{dx}[\sin x] = \cos x\)
- Exponential: \(\frac{d}{dx}[e^x] = e^x\)
Worked Examples
Trigonometric
\(\frac{d}{dx}[\sin x] = \cos x\)
Exponential
\(\frac{d}{dx}[e^x] = e^x\)
External educational resource
Open the GeoGebra graphing calculator
Want to graph a function we don't simplify symbolically (e.g. tan, products, compositions)? GeoGebra's free Graphing Calculator handles it and lets you drag points along the curve.
GeoGebra (geogebra.org)