Basics of Derivative Calculus with its Explanation and Solved Examples

In calculus, a derivative is a fundamental concept in Mathematics that describes the rate at which the function changes its behavior concerning a given value. It provides information about how a function behaves locally near a particular point.

The concept of the derivative is closely related to integration, as they are inverse operations of each other. Integration allows us to find the original function when given its derivative, while differentiation allows us to find the derivative when given the original function.

In this article, we will discuss the definition of a derivative, its basic concept, its application of derivative, Advantages, and disadvantages of a derivative. Also, we explain the derivative with the help of an example.

Definition

A derivative is the concept of Mathematics that the function changes the behavior for the given value. Derivatives have widespread applications in many areas of science, engineering, and mathematics. They are used to analyze and model the behavior of functions, optimize functions, solve differential equations, and understand rates of change in various real-world phenomena.

Basic Concepts of Derivatives

The basic concept we first understand by using the derivative of any function. If we know some basic keys of the derivative with the help of this, we can find easily the derivative of any function.

• Slope: Derivatives represent the slope of a function at a particular point. They indicate the steepness of the curve or line that represents the function at that point.
• Tangent Line: The derivative of a function at a given point gives the slope of the tangent line to the curve at that point. The instantaneous rate of change of the function at that location is shown by the tangent line.
• Instantaneous Rate: Derivatives allow us to determine the instantaneous rate of change of a function at a specific point. This provides insight into the behavior of the function at that precise moment.
• Notation: The common way to represent a function’s derivative is as f'(x) or dy/dx, where dy stands for the change in the function’s output and dx stands for the change in its input.

Application of derivative

The derivative is a fundamental concept in calculus and has numerous applications across various fields. Some of the common applications of derivatives include:

• Calculating rates of change:

Derivatives are used to measure how one quantity changes with respect to another. For example, in physics, derivatives can be used to determine the velocity of an object by taking the derivative of its position with respect to time.

• Physics and motion:

Derivatives are extensively used in physics to study the motion of objects. For instance, the derivative of displacement with respect to time gives the velocity, and the derivative of velocity with respect to time gives the acceleration.

• Economic analysis:

Derivatives play a crucial role in economics, especially in the study of marginal analysis. Marginal cost, marginal revenue, and marginal utility are all concepts that involve derivatives and help analyze the change in costs, revenues, and utility with respect to the change in quantity.

• Engineering and science:

Derivatives are applied in various branches of engineering and science, including electrical engineering, chemical engineering, and biology. They are used to analyze and optimize systems, model physical phenomena, and understand complex relationships between variables.

• Computer graphics:

Derivatives find extensive use in computer graphics for rendering, shading, and simulating the behavior of objects in virtual environments.

Advantages and disadvantages of derivative

Advantages of derivatives:

• Rate of Change Analysis:Derivatives provide a quantitative measure of how a function changes, allowing for precise analysis of rates of change. This is valuable in various fields such as physics, economics, and engineering.
• Optimization:In optimization issues, where the objective is to determine the maximum or least value of a function, derivatives are crucial. Derivatives help identify critical points and determine whether they correspond to local maxima or minima.
• Physics Applications:In physics, derivatives are essential because they are used to model dynamic systems, compute velocities and accelerations, and characterize the motion of objects.

Disadvantages of derivatives:

• Limitations in Discontinuous Functions:Derivatives may not exist or behave as expected in functions with discontinuities or sharp changes. Discontinuities pose challenges in calculating derivatives, and alternative approaches such as generalized derivatives may be required.
• Complex Functions:Derivatives become more challenging to compute as functions become more complex, involving multiple variables, composite functions, or higher orders of derivatives. Analytical solutions may not always be readily available, requiring numerical methods or approximations.
• Interpretation and Context:Derivatives provide quantitative information about a function’s rate of change, but their interpretation and application require careful consideration of the context and underlying assumptions. Misinterpretation or misuse of derivatives can lead to erroneous conclusions or decisions.

Examples of Derivative:

Example No.1:

f(x) = 3x² – 2x + 1.

d/d(x)[f(x)] =?

Solution:

Step 1:

Identify the power rule for derivatives, which states that if we have a term of the form axⁿ the derivative is given by d/dx(axⁿ) = nax^(n-1).

Step 2:

Apply the power rule to each term of the function.

• For the term 3x², the derivative is d/dx(3x^2) = 2(3)x^(2-1) = 6x.
• For the term -2x, the derivative is d/dx(-2x) = (-2) (1)x^(1-1) = -2.
• For the constant term 1, the derivative is zero since the derivative of a constant is always zero.

Step 3:

Combine the derivatives of each term to obtain the derivative of the function.

• d/d(x)[3x²] = 6x.
• d/d(x)[-2x] = -2.
• d/d(x)[1] =0

Therefore, the derivative of f(x) = 3x² – 2x + 1 is

f'(x) = 6x – 2.

A derivative calculator with steps can be used as an alternative to solve the problems of the differential calculus in a step by step manner in seconds.

Example 2:

P(x)=sin(x)/cos(x)

d/d(x)[P(x)] =?

Solution:

find the derivative of a given function

P(x)=[sin(x)/cos(x)]

Step 1:

The quotient rule is given:

d/dx[f(x)g(x)] =[−f(x)d/dx(g(x)) +g(x)d/dx f(x)]/g²(x)

Step 2:

In the given question

P(x)=sin(x) and g(x)=cos(x)

To find d/dx(f(x))

The derivative of sin is cos:

d/dx(sin(x)) =cos(x)

To find d/dx(g(x))

d/dx(cos(x)) =−sin(x)

Step 3:

According to the quotient rule we have:

sin²(x)+cos²(x)/cos²(x)

Now simplify:

1/cos²(x)

The answer is:

P^’(x)=1/cos²(x)

Conclusion

In this article, we have discussed the definition of a derivative, its basic concept, application of derivative, and Advantages and disadvantages of derivative. Also, we explain the derivative with the help of an example. Anyone can easily represent this topic after completing studying this article.