Derivative

Derivative is a term used in mathematics, specifically in calculus, to describe the rate at which a quantity changes. It is a fundamental concept in the field of calculus and has wide applications in various fields such as physics, engineering, and economics.

Definition

In calculus, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.

Applications

Derivatives have wide applications in various fields. In physics, they are used to determine the velocity and acceleration of a moving object. In engineering, they are used in the design of systems and controls. In economics, they are used to find the maximum and minimum values of functions, including cost and profit.

Types of Derivatives

There are several types of derivatives, including:

• Partial Derivative: This is the derivative of a function of multiple variables when all but the variable of interest are held fixed during the differentiation.

• Directional Derivative: This measures the rate at which the function changes at a point in the direction of one of its arguments.

• Second Derivative: This is the derivative of the derivative of a function. It can be used to determine whether a point is a local maximum or minimum.

Calculating Derivatives

The derivative of a function 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.

See Also

• Calculus
• Function (mathematics)
• Limit (mathematics)

References

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