RUBIK'S CUBE PRESENTATION

Derivative and differential of functions in a point

Conclusion

Properties of the differential of a function at a point

Properties of the derivative of a function at a point

Differential of a function at a point

Derivative of a function at a point

Introduction

Plan

Introduction

Derivatives and differentials of functions are key concepts in mathematical analysis and are widely used in many areas of science and technology, such as physics, economics, finance, engineering, etc

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Derivative and differential are concepts used in calculus to study the behavior of functions.

Introduction

Derivative of a function at a point

The geometric meaning of the derivative of a function at a point is that the derivative f'(x0) determines the tangent of the slope of the tangent to the graph

Let f(x) be a function of one variable, then the derivative of the function f(x) at the point x0 is defined as the limit of the ratio of the increment of the function to the increment of the argument as the increment of the argument tends to zero: f'(x0) = lim (f(x) - f(x0))/(x - x0), x -> x0

Derivative of a function at a point

Differential of a function at a point

The geometric meaning of the differential is that it is tangent to the curve given graphically by a function at a given point.

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The differential of a function at a point is a linear approximation to the increment of the function at that point. If the function f(x) has a differential at the point x = a, then this differential is denoted as df(x) or df(a) and is defined as follows:

Differential of a function at a point

From this example, we used the derivative of the function to calculate the value of the differential at a given point. Then we got a linear approximation to the change of the function with small changes in the argument at this point.

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Calculate the differential of the function f(x) = x^2 at the point x = 2.

Example 1:

Properties of the derivative of a function at a point

4. Derivative of an inverse function:

3. Derivative of a composite function:

2. Derivative of a sum, difference, product, and quotient of functions:

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1. Linearity of the derivative function:

Properties of the derivative of a function at a point

Properties of the differential of a function at a point

4. Inverse function differential:

3. Differential of a complex function (rule of differentiation of a complex function):

2. Differential of the sum, difference, product and quotient of functions:

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1. Linearity of the function differential at a point:

Properties of the differential of a function at a point

Conclusion

Conclusion

Derivatives and differentials are important tools of mathematical analysis and are widely used in many fields of science, technology and economics.