Differentiable function

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Differentiable function

A differentiable function is a function that possesses a derivative at each point in its domain. Differentiability implies continuity, but the converse is not necessarily true. Differentiable functions are a central concept in calculus and mathematical analysis.

Definition

A function \( f: \mathbb{R} \to \mathbb{R} \) is said to be differentiable at a point \( a \in \mathbb{R} \) if the limit \[ \lim_Template:H \to 0 \fracTemplate:F(a+h) - f(a){h} \] exists. This limit, if it exists, is called the derivative of \( f \) at \( a \) and is denoted by \( f'(a) \) or \( \frac{df}{dx}(a) \).

Properties

• **Continuity**: If a function is differentiable at a point, it is also continuous at that point.
• **Linearity**: The derivative of a sum of functions is the sum of the derivatives.
• **Product Rule**: The derivative of a product of two functions is given by \( (fg)' = f'g + fg' \).
• **Quotient Rule**: The derivative of a quotient of two functions is given by \( \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2} \).
• **Chain Rule**: The derivative of a composite function is given by \( (f \circ g)' = (f' \circ g) \cdot g' \).

Higher-Order Derivatives

A function is said to be twice differentiable if its derivative is also differentiable. The second derivative is denoted by \( f(x) \) or \( \frac{d^2f}{dx^2} \). Higher-order derivatives can be defined similarly.

Differentiability in Higher Dimensions

For functions of several variables, differentiability is defined in terms of partial derivatives. A function \( f: \mathbb{R}^n \to \mathbb{R} \) is differentiable at a point \( \mathbf{a} \in \mathbb{R}^n \) if there exists a linear map \( L: \mathbb{R}^n \to \mathbb{R} \) such that \[ \lim_{{\mathbf{h} \to \mathbf{0}}} \frac{{f(\mathbf{a} + \mathbf{h}) - f(\mathbf{a}) - L(\mathbf{h})}}{\|\mathbf{h}\|} = 0. \] The map \( L \) is called the Jacobian matrix of \( f \) at \( \mathbf{a} \).

Examples

• The function \( f(x) = x^2 \) is differentiable everywhere, and its derivative is \( f'(x) = 2x \).
• The function \( f(x) = |x| \) is not differentiable at \( x = 0 \) because the left-hand and right-hand limits of the difference quotient are not equal.

Related Concepts

• Continuous function
• Derivative
• Partial derivative
• Jacobian matrix
• Chain rule
• Product rule
• Quotient rule

See Also

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

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