Derivatives: Formula, Types, Applications and Examples

The derivative of a function at a point is the rate of change or slope of the tangent line to the function at that given point.

• Derivatives are essential for solving calculus and differential equation problems.
• Scientists observe changing systems to determine the rate of change of a specific variable.
• They then convert this information into a differential equation.
• Finally, by using integration, they derive a function that predicts the behavior of the system under different conditions.
• The derivative is used to determine the sensitivity of one variable (dependent variable) to another (independent variable).

The derivative of function y with respect to x can be represented by

\(\frac {dy}{dx}\)

Where dx represents an infinitesimal change in x

Also Read: Applications of Derivatives

Key Terms: Intervals, Derivatives, Function, Rate of change, Constant, Slope of a tangent line, Velocity, Instantaneous acceleration, Differentiation, Independent variable, anti-differentiation

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Derivatives

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In mathematics, derivatives relate to the instantaneous rate of change of one quantity with respect to another.

• It is beneficial to study the nature of an amount at any given time.
• The process of determining the derivative is known as differentiation.
• The reverse process is known as anti-differentiation.

The derivative of a function f(x) = y with respect to the variable x is represented by dy/dx or f'(x) and is given by

\(\frac {dy}{dx}=lim_{h \rightarrow0}\frac{f(x+h)-f(x)}{h}\)

Let us suppose that a function calculates the rate of change in velocity, or, as we know it, the acceleration of a vehicle as it goes from one place to another.

• The rate of change in velocity is instantaneous because the function is reliant on both the vehicle's speed and direction.
• To determine the vehicle's instantaneous acceleration, we must first determine the function's limitations at that time.
• A derivative of another function is a function that denotes the rate of change of that function.
• In other words, a derivative is used to define a function's rate of change.

Also Read: Differential Equations

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Examples of Derivatives

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Let’s discuss an example of derivatives step by step below:

1. Find the derivative of the below equation using the derivative:
2. So, what we need to do is insert this function into the derivative formulation and apply some mathematics. While the algebra will, admittedly, be quite unpleasant at times, it's only algebra, so don't get too enthusiastic about the fact that we're now computing derivatives.

First, enter the function into the derivative definition:

1. We know that we can't just enter h = 0 since this would result in a division by zero mistake. So we'll have to put forth some effort. In this situation, multiplying everything out and spreading the negative sign through in the second term is required. This results in,
2. Every term in the numerator that did not include an h canceled out, and we can now factor an h out of the numerator, which will cancel out the h in the denominator. The limit may then be computed.

Also Read: Continuity and Differentiability

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Interpretation of Derivatives

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There are methods to interpret the derivatives. Such methods are explained below:

• Rate of change: Derivatives are used to calculate the rate of change of one item in relation to another. We may discover the estimated change in one quantity with regard to the change in the other quantity by utilizing derivatives. Assuming we have a function y = f(x) defined in the interval [a, a+h], the average rate of change in the function in the provided interval is

(f(a + h)-f(a))/h

Now by the derivative definition, we can write

That is the instant rate of change of the function f(x) at a.

Now, for small value of h, we can write

f'(a) ≈ (f(a+h) − f(a))/h

• Approximation value: A function's derivative can be used to get the linear approximation of a function at a given value. Newton proposed the linear approximation technique, which entails first determining the value of the function at the given location and then determining the equation of the tangent line to determine the estimated close value to the function.
• The slope of a tangent line: If we have a function curve and wish to discover the equation of the tangent to a curve at a particular location, we can use the derivative to calculate the slope and equation of the tangent line. A tangent is a line to a curve that only touches the curve at one point and has a slope equal to the curve's derivative at that location.
• We can determine the equation of the tangent line to the curve by calculating the slope of the tangent line to the curve and applying the equation. Similarly, we may obtain the equation of the normal line to a function's curve at a given location. This normal line will be perpendicular (normal) to the tangent line.

Also Read: Approximations

Limits and Derivatives Detailed Video Explanation

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Formula of Derivatives

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1] DERIVATIVE FORMULAS FOR ELEMENTARY FUNCTIONS

• d/dx (k) = 0, where k is any constant
• d/dx(x) = 1
• d/dx(xn) = nx^n-1
• d/dx (kx) = k, where k is any constant
• d/dx (√x) = 1/2√x
• d/dx (1/x) = -1/x²
• d/dx (log x) = 1/x, x > 0
• d/dx (ex) = ex
• d/dx (ax) = ax log a

2] DERIVATIVE FORMULAS FOR TRIGONOMETRIC FUNCTIONS

We may also compute the derivative of trigonometric functions, such as sin, cos, and tan. The formulae are as follows:

• d/dx (sin x) = cos x
• d/dx (cos x) = -sin x
• d/dx (tan x) = sec²x
• d/dx (cosec x) = -cosec x cot x
• d/dx (sec x) = sec x tan x
• d/dx (cot x) = -cosec²

^​Also Read: tangents and normals

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Types of Derivatives

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Derivatives are categorised according to their order, such as first and second order derivatives. These are defined further down.

1. Derivative of First-Order: The first order derivatives reveal the function's direction, whether it is rising or decreasing. The first derivative math, also known as the first-order derivative, may be seen as an instantaneous rate of change. The slope of the tangent line can also be used to forecast it.
2. Derivative of Second-Order: Second-order derivatives are used to determine the form of the graph for a particular function. The functions can be categorised based on their concavity. The provided graph function's concavity is categorised into two types- Concave up & Concave Down.

Also Read: increasing and decreasing functions

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Derivative Notations

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Different mathematicians have developed a variety of notations for the derivative of a function depending on the context in which they are employed. The following are some examples of common notations:

1. Leibniz's notation is applied when a functional connection between the dependent and independent variables y and x produces the equation y=f(x). 
2. Lagrange's notation -In Lagrange's notation, a prime mark is used to represent a derivative. is the prime mark for the first derivative. As a result, the strength of the subsequent derivatives continues to grow.
3. Euler's Notation -In Euler's notation, a differential operator is proposed. When applied to a function of x, the derivative.
4. Dot notation is another name for Newton's notation. A dot is placed on the dependent variable.

Also Read: Differentiation and Integration Formula

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Applications of Derivatives

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Derivatives have a wide range of applications, not just in mathematics but also in everyday life. for example, have many significant uses in mathematics, such as determining the rate of change of a quantity, determining the approximation value, determining the equation of Tangent and Normal to a Curve, and determining the Minimum and Maximum Values of algebraic expressions. Derivatives are widely used in disciplines such as science, engineering, physics, and so on. The most prevalent use of derivatives may be found in:

• Calculating the Rate of Change of a Quantity
• Obtaining an Approximation Value
• Finding the Tangent and Normal to a Curve Equation
• Finding the Maxima and Minima, as well as the Point of Inflection
• Identifying the Increasing and Decreasing Functions

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Things to Remember

• The derivative value indicates the direction of a curve at a specific point.
• The derivative value at each point on the graph is the slope of the tangent line at that point.
• The exact derivative formula is ddx. xn=n. xn−1 d d x.
• Derivatives relate to the instant rate of change of one quantity with relation to another. It is beneficial to explore the nature of a quantity on a moment-to-moment basis.

Also Read:

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