Derivative Calculator
Compute derivatives of functions with respect to variables using symbolic differentiation rules. Get step-by-step solutions to understand how to differentiate any function.
Derivative Calculation Results
Function Plot
Enter a function to see its plot and derivative
Original function: \( f(x) = x^2 + \sin(x) \)
Derivative: \( \frac{d}{dx}\left(x^2 + \sin(x)\right) = 2x + \cos(x) \)
Step-by-Step Solution
Using the power rule: \( \frac{d}{dx} x^n = n x^{n-1} \)
\( \frac{d}{dx} x^2 = 2x \)
Derivative of sine is cosine: \( \frac{d}{dx} \sin(x) = \cos(x) \)
Sum rule: \( \frac{d}{dx} (f(x) + g(x)) = \frac{d}{dx} f(x) + \frac{d}{dx} g(x) \)
\( \frac{d}{dx}\left(x^2 + \sin(x)\right) = 2x + \cos(x) \)
About Derivatives
The derivative of a function represents the rate at which the function's value changes with respect to changes in its input variable. In geometric terms, the derivative at a point equals the slope of the tangent line to the function's graph at that point.
Basic Differentiation Rules
Several fundamental rules make calculating derivatives straightforward:
- Power Rule: \( \frac{d}{dx} x^n = n x^{n-1} \)
- Constant Rule: \( \frac{d}{dx} c = 0 \) (where c is a constant)
- Sum Rule: \( \frac{d}{dx} [f(x) + g(x)] = \frac{d}{dx} f(x) + \frac{d}{dx} g(x) \)
- Product Rule: \( \frac{d}{dx} [f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x) \)
- Quotient Rule: \( \frac{d}{dx} \left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \)
- Chain Rule: \( \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \)
Common Derivatives
Memorizing derivatives of common functions is essential:
| Function | Derivative |
|---|---|
| \( \sin(x) \) | \( \cos(x) \) |
| \( \cos(x) \) | \( -\sin(x) \) |
| \( \tan(x) \) | \( \sec^2(x) \) |
| \( e^x \) | \( e^x \) |
| \( \ln(x) \) | \( \frac{1}{x} \) |
| \( a^x \) | \( a^x \ln(a) \) |
Applications of Derivatives
Derivatives have numerous applications across mathematics and sciences:
- Physics: Velocity is the derivative of position with respect to time; acceleration is the derivative of velocity.
- Economics: Marginal cost is the derivative of the total cost function.
- Biology: Population growth rates can be modeled with derivatives.
- Engineering: Derivatives help in optimizing designs and analyzing systems.
- Graphing: Derivatives help identify maxima, minima, and inflection points.
Higher-Order Derivatives
The second derivative measures how the rate of change of a quantity is itself changing. It can indicate acceleration in physics or concavity in mathematics. Higher-order derivatives (third, fourth, etc.) are found by repeatedly differentiating the function.
Notation for higher-order derivatives:
- First derivative: \( f'(x) \) or \( \frac{df}{dx} \)
- Second derivative: \( f''(x) \) or \( \frac{d^2f}{dx^2} \)
- n-th derivative: \( f^{(n)}(x) \) or \( \frac{d^nf}{dx^n} \)