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Quick Answer
The derivative of x^2 + 3x is 2x + 3. At x = 2, the derivative equals 7.
Use * for multiplication. Examples: x^2, sin(x), exp(x), log(x), sqrt(x)
Common Examples
| Input | Result |
|---|---|
| f(x) = x^2 | f'(x) = 2x |
| f(x) = x^3 + 2x | f'(x) = 3x^2 + 2 |
| f(x) = sin(x) | f'(x) = cos(x) |
| f(x) = e^x | f'(x) = e^x |
| f(x) = ln(x) | f'(x) = 1/x |
How It Works
Definition
The derivative of f(x) is defined as:
\[f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\]This measures the instantaneous rate of change of the function at any point.
Common differentiation rules
| Function | Derivative |
|---|---|
| \(x^n\) | \(n x^{n-1}\) (power rule) |
| \(\sin(x)\) | \(\cos(x)\) |
| \(\cos(x)\) | \(-\sin(x)\) |
| \(e^x\) | \(e^x\) |
| \(\ln(x)\) | \(1/x\) |
| \(f(g(x))\) | \(f'(g(x)) \cdot g'(x)\) (chain rule) |
Sum rule: \((f + g)' = f' + g'\)
Product rule: \((fg)' = f'g + fg'\)
Quotient rule: \((f/g)' = (f'g - fg') / g^2\)
Worked example
For f(x) = x^3 + 2x: Apply the power rule to each term. The derivative of x^3 is 3x^2 (bring down the exponent, subtract 1). The derivative of 2x is 2. So f’(x) = 3x^2 + 2. At x = 1: f’(1) = 3(1)^2 + 2 = 5, meaning the slope of the function at x = 1 is 5.
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Frequently Asked Questions
What does the derivative represent?
The derivative of a function at a point gives the slope of the tangent line at that point. It represents the instantaneous rate of change. If f(x) represents position over time, f'(x) represents velocity. If f(x) represents cost as a function of quantity, f'(x) represents marginal cost.
What notation is used for derivatives?
Common notations include f'(x) (Lagrange notation), df/dx (Leibniz notation), and Df (operator notation). For higher-order derivatives: f''(x) or d^2f/dx^2 for the second derivative, and so on. This calculator uses Lagrange notation (f'(x)).
What functions can this calculator differentiate?
This calculator handles polynomials, trigonometric functions (sin, cos, tan), exponential functions (exp, e^x), logarithms (log for natural log), square roots (sqrt), and combinations using addition, subtraction, multiplication, division, and function composition.
What if the derivative does not exist at a point?
A function may not be differentiable at points where it has a sharp corner (like |x| at x=0), a vertical tangent, or a discontinuity. If you try to evaluate the derivative at such a point, the result may be undefined or infinite.
How is the numeric derivative calculated?
When symbolic differentiation is not possible, a numeric approximation is used via the central difference method: f'(x) is approximately [f(x+h) - f(x-h)] / (2h) with a very small h. This gives accurate results for smooth functions but may have small rounding errors.