Derivative Calculator
Calculate derivatives and find the slope of functions at any point
Use: x^2, sqrt(x), sin(x), cos(x), tan(x), ln(x), e^x, pi
The x-value where you want to find the derivative (slope)
Smaller step size = higher accuracy but may cause numerical errors for very small values
f'() = d/dx [f(x)] =
Slope of tangent line at x = Derivative of f(x)
Calculation Details
Power Functions
(d/dx)[xⁿ] = n·xⁿ⁻¹
(d/dx)[1/x] = -1/x²
(d/dx)[√x] = 1/(2√x)
Trigonometric
(d/dx)[sin(x)] = cos(x)
(d/dx)[cos(x)] = -sin(x)
(d/dx)[tan(x)] = sec²(x)
Exponential & Log
(d/dx)[eˣ] = eˣ
(d/dx)[aˣ] = aˣ·ln(a)
(d/dx)[ln(x)] = 1/x
Differentiation Rules
(d/dx)[k·f] = k·f'
(d/dx)[f+g] = f' + g'
(d/dx)[f·g] = f'g + fg'
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About Derivative Calculator
What is a Derivative?
A derivative measures how a function changes as its input changes. It represents the instantaneous rate of change or the slope of the tangent line to a curve at any given point. Derivatives are fundamental to calculus and have applications in physics, engineering, economics, and many other fields.
How to Use This Calculator
- Select Calculation Type: Choose between finding the derivative at a point (numerical) or symbolic differentiation
- Enter Function: Type your function using standard mathematical notation
- Enter Point (for numerical): Specify the x-value where you want to find the derivative
- Calculate: Click calculate to get your result
Understanding Your Results
Derivative at a Point
The derivative at a specific point gives you the slope of the tangent line at that point. This is a numerical value mapping to the instantaneous rate of change.
Symbolic Derivative
A symbolic derivative finds the general derivative function. The result is a formula showing the rate of change for any input x.
Numerical Method: Central Difference
This calculator uses the central difference method for numerical differentiation, which provides better accuracy than forward or backward differences:
Formula: f'(x) ≈ (f(x+h) - f(x-h))/(2h)
Where h is a small step size (default: 0.0001).
Common Derivative Formulas
| Function | Derivative |
|---|---|
| xⁿ | n·xⁿ⁻¹ |
| eˣ | eˣ |
| ln(x) | 1/x |
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| tan(x) | sec²(x) |
Frequently Asked Questions (FAQ)
What is a good way to check derivative results?
You can verify numerical derivatives by comparing them with symbolic derivatives evaluated at the same point.
How accurate is the numerical derivative?
The central difference method with a small step size provides precision to about 8 decimal places for smooth functions.
Differentiation Rules
Power Rule
(d/dx)[xⁿ] = n·xⁿ⁻¹
Constant Multiple Rule
(d/dx)[k·f(x)] = k·f'(x)
Sum Rule
(d/dx)[f(x) + g(x)] = f'(x) + g'(x)
Disclaimer: This calculator is for educational purposes only. For complex engineering or scientific applications, verify results with professional mathematics software.