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derivative
Also known as derivative (function), differentiation, instantaneous rate of change
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. The derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a d
AI overview
A derivative measures how sensitive a function is to changes in its input—essentially telling you how quickly the output changes at any given point. It matters because it's a fundamental mathematical tool that helps us understand rates of change in everything from physics to economics, and it provides the best straight-line approximation of how a function behaves near a specific point.
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Described at
Derivatives - Calculus, Meaning, Interpretation
A derivative is the rate of change of a function with respect to a variable. The derivative of a function f(x) is denoted by f'(x) and it can be found by using the limit definition lim h→0 (f(x+h)-f(x))/h.
cuemath.com →Let us learn what exactly a derivative means in calculus and how to find it along with rules and examples. Thus, whenever we see the phrases like "slope/gradient", "rate of change", "velocity (given the displacement)", "maximize/minimize" etc then it means that the concept of derivatives is involved. The three basic derivatives of the algebraic , logarithmic / exponential and trigonometric functions are derived from the first principle of differentiation and are used as standard derivative formulas . They are as follows. Further, we can find the second-order partial derivatives also like ∂2f/∂x2, ∂2f/∂y2, ∂2f/∂x ∂y, and ∂2f/∂y ∂x. The concept of slope, and hence the derivatives, is used to find the maximum or minimum value of a function. There are two tests that use derivatives and are used to find the maxima/minima of a function. They are The second derivative test uses the critical points and the second derivative to find the maxima/minima. To perform this test: Since the velocity is positive, the object is moving to the right side. Indulging in rote learning, you are likely to forget concepts. With Cuemath, you will learn visually and be surprised by the outcomes.
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~25 min read
Article
21 sectionsContents
- Definition
- As a limit
- Using infinitesimals
- Continuity and differentiability
- Notation
- Rules of computation
- Rules for basic functions
- Rules for combined functions <span class="anchor" id="Rules"></span>
- Computation example
- Antidifferentiation
- Higher-order derivatives<span class="anchor" id="order of derivation"></span><span class="anchor" id="Order"></span>
- In other dimensions
- Vector-valued functions
- Partial derivatives
- Directional derivatives
- Total derivative and Jacobian matrix
- Generalizations
- See also
- Notes
- References
- External links
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. The derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation.
There are multiple different notations for differentiation. Leibniz notation, named after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas prime notation is written by adding a prime mark. Higher order notations represent repeated differentiation, and they are usually denoted in Leibniz notation by adding superscripts to the differentials, and in prime notation by adding additional prime marks. Higher order derivatives are used in physics; for example, the first derivative with respect to time of the position of a moving object is its velocity, and the second derivative is its acceleration.