Differentiable function
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[edit | edit source]
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[edit | edit source]
- **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[edit | edit source]
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[edit | edit source]
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[edit | edit source]
- 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[edit | edit source]
- Continuous function
- Derivative
- Partial derivative
- Jacobian matrix
- Chain rule
- Product rule
- Quotient rule
See Also[edit | edit source]
Part of a series of articles about |
Calculus |
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<math>\int_{a}^{b} f'(t) \, dt = f(b) - f(a)</math> |
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