Derivation

Mathematical concept of deriving functions

For other uses, see Derivation (disambiguation).

Derivation in mathematics refers to the process of finding the derivative of a function. The derivative measures how a function changes as its input changes. Derivation is a fundamental tool in calculus and is used to solve problems in a wide range of fields including physics, engineering, economics, and biology.

Definition

The derivative of a function \( f \) at a point \( x \) is defined as the limit: \[ f'(x) = \lim_Template:H \to 0 \fracTemplate:F(x+h) - f(x){h} \] if this limit exists. The function \( f' \) is called the derivative of \( f \).

Notation

Several notations are commonly used to denote the derivative of a function. These include:

• Leibniz notation: \(\frac{df}{dx}\)
• Lagrange notation: \(f'(x)\)
• Newton notation: \(\dot{f}\) (used primarily in physics)
• Euler notation: \(D_x f\)

Rules of Derivation

There are several important rules for finding derivatives, including:

• The power rule: \(\frac{d}{dx} x^n = nx^{n-1}\)
• The product rule: \(\frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)\)
• The quotient rule: \(\frac{d}{dx} \left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}\)
• The chain rule: \(\frac{d}{dx} f(g(x)) = f'(g(x))g'(x)\)

Applications

Derivatives have numerous applications in various fields:

• In physics, derivatives describe the rate of change of physical quantities, such as velocity and acceleration.
• In economics, derivatives are used to find marginal cost and marginal revenue.
• In biology, derivatives can model population growth rates and other dynamic processes.

Higher-Order Derivatives

The second derivative of a function is the derivative of the derivative, denoted as \( f(x) \) or \(\frac{d^2f}{dx^2}\). Higher-order derivatives can be found similarly and are used in various applications, such as in the study of concavity and inflection points.

Related Concepts

• Integral
• Differential equation
• Partial derivative
• Gradient
• Tangent line

See Also

• Calculus
• Mathematical analysis
• Function (mathematics)
• Limit (mathematics)
• Continuity (mathematics)

Related Pages

• Derivative
• Integral
• Calculus
• Differential equation
• Partial derivative
• Gradient