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 |
---|
<math>\int_{a}^{b} f'(t) \, dt = f(b) - f(a)</math> |
Navigation: Wellness - Encyclopedia - Health topics - Disease Index - Drugs - World Directory - Gray's Anatomy - Keto diet - Recipes
Search WikiMD
Ad.Tired of being Overweight? Try W8MD's physician weight loss program.
Semaglutide (Ozempic / Wegovy and Tirzepatide (Mounjaro / Zepbound) available.
Advertise on WikiMD
WikiMD is not a substitute for professional medical advice. See full disclaimer.
Credits:Most images are courtesy of Wikimedia commons, and templates Wikipedia, licensed under CC BY SA or similar.
Translate this page: - East Asian
中文,
日本,
한국어,
South Asian
हिन्दी,
தமிழ்,
తెలుగు,
Urdu,
ಕನ್ನಡ,
Southeast Asian
Indonesian,
Vietnamese,
Thai,
မြန်မာဘာသာ,
বাংলা
European
español,
Deutsch,
français,
Greek,
português do Brasil,
polski,
română,
русский,
Nederlands,
norsk,
svenska,
suomi,
Italian
Middle Eastern & African
عربى,
Turkish,
Persian,
Hebrew,
Afrikaans,
isiZulu,
Kiswahili,
Other
Bulgarian,
Hungarian,
Czech,
Swedish,
മലയാളം,
मराठी,
ਪੰਜਾਬੀ,
ગુજરાતી,
Portuguese,
Ukrainian
Contributors: Prab R. Tumpati, MD