Derivative calculus is an important and fundamental dimension of the calculus. Derivative provides us with a fundamental structure for modeling, explaining, and predicting real-world problems. It was established in the 19th century and now it is an important branch of mathematics, which plays a significant role in the development of advanced and modern science.

Sir Isaac Newton and Gottfried independently developed the idea of ​​a derivative function, sometimes called calculus in the 17th century. Although both of them arrived at a similar concept for the derivation, they approached the problem in different ways.

Newton considered variables having the function of time; while Leibniz gave an idea of ​​the variables x and y implying a sequence of infinitely close values. In this blog, we will elaborate on the concept of derivatives along with its types as well as we will solve some of its important examples.

What is a Derivative?

In mathematics, the derivative is the rate of change of a function with respect to a variable. The commonly used symbol for the derivative is f’(x) (read as f prime x) or df/dx (read as a derivative of “f” w.r.t. x). Derivatives play a fundamental role in solving problems in calculus and differential equations.

Geometrically, the derivative of a function is the slope of the graph of the function or, more precisely, the slope of the tangent line at a point, which can be computed using a given relation.

Slope = m = 𝚫y/𝚫x = (y – y₁)/ (x – x₁)

Types of Derivatives:

Here we will describe some of the important types of derivatives.

Partial Derivatives:

The measure of the rate of change of a multivariable function w.r.t. one of its variables treating all other variables of this multivariable function constant is known as partial derivative. Let us have a function f(x,y), then its partial derivatives are

• fx = ∂f(x,y)/ ∂x, a partial derivative of f with respect to x.
• fy = ∂f(x,y)/ ∂y, the partial derivative of f with respect to y.

The partial derivative is very important to measure and observe the impact of one variable on the functional behavior in the case of a multivariable function.

Implicit Derivatives:

The derivative of an implicit function is termed an implicit derivative. It is obvious that the dependent variable can’t be expressed explicitly in terms of the independent variable in an implicit function.

So, when we take the derivative of a dependent variable w.r.t. independent variable appearing implicitly in the equation, then it is known as implicit differentiation.

Implicit differentiation is useful in solving problems related to differential equations, integral equations, calculus, physics, etc. It illustrates precisely while finding and analyzing surfaces, curves, and geometrical shapes of different functions defined implicitly.

Explicit derivatives:

When we take the derivative of an explicit function it is called an explicit derivative. The concept of explicit differentiation is very useful in the study of calculus, mathematical analysis, and physics to apprehend optimization problems, slope of curves, etc.

Total Derivatives:

The total mensuration of the rate of change of a multivariable function w.r.t all of its independent variables manifesting in the function is termed the total derivative of the function.

Suppose we have a function y = g(x, y, z) of three variables where y is the dependent variable and x, y, and z are independent variables, the total derivative of g w.r.t. x, y, and z will be the total change in g due to a change in both x, y and z. mathematically,

dg = (∂g/ ∂x) dx + (∂g/ ∂y) dy + (∂g/ ∂z) dz.

How to find the derivative of a function?

Follow the below examples to learn how to find the derivative of a function.

Example 1:

Compute dy/dx for 3x² – y = 5x.

Solution:

Step 1: Given data

3x² – y = 5x

Step 2: Rearrange the above equation to write y explicitly.

y = 3x² – 5x

Step 3: Differentiate w.r.t x.

dy/dx = 3(2x) – 5

dy/dx = 6x – 5 which is the required answer.

Example 2:

Compute dy/dx for the given equation by implicit differentiation.

2x³ + y² = x²y

Solution:

Step 1: Given data

2x³ + y² = x²y                     (1)

Step 2: Differentiating (1) on both sides w.r.t x.

d/dx (2x³ + y²) = d/dx (x²y)

Step 3: Simplify

6x² + (2y)dy/dx = 2xy + x²dy/dx

(2y) dy/dx – x² dy/dx = 2xy – 6x²

(2y – x²) dy/dx = 2xy – 6x²

dy/dx = (2xy – 6x²)/ (y – x²) which is the required answer.

Example 3:

Find out the partial derivative of the function g(x, y) = 2x⁴y² with respect to x and y.

Solution:

Step 1: Write down the given function.

g(x, y) = 2x⁴y² ———–(1)

Step 2: Differentiating (1) with respect to x.

∂g/∂x = ∂/∂x (2x⁴y²)

Step 3: Here we will treat ‘y’ as a constant term.

∂g/∂x = 2(4x³) y²

∂g/∂x = 8x³y² which is the partial derivative of ‘g’ w.r.t x.

Step 4: Differentiating (1) with respect to ‘y’.

∂g/∂y = ∂/∂y (2x⁴y²)

Step 5: here we will treat ‘x’ as a constant term.

∂g/∂y = 2x⁴(2y)

∂g/∂y = 4x⁴y which is the partial derivative of ‘g’ w.r.t ‘y’.

So,

∂g/∂x = 8x³ y2 and ∂g/∂y = 4x⁴ y Ans.

Example 4:

Find the total derivative of the function f(x, y) = 3x⁴ + 2y³

Solution:

Step 1: The given function is

f(x, y) = 3x⁴ + 2y³                (1)

Step 2: Differentiating (1) w.r.t x and treating y as a constant.

∂f/∂x = 3(4x³) + 0

∂f/∂x = 12x³

Step 2: Differentiating (1) w.r.t y and treat x as a constant.

∂f/∂y = 0 + 2(3y²)

∂f/∂y = 6y²

Step 3: Now sum up the values of ∂f/∂x and ∂f/∂y to find out the total derivative of the given function.

dy/dx = ∂f/∂x + ∂f/∂y

dy/dx = 12x³ + 6y² which is s the required answer.

Conclusion:

In this blog, we’ve discussed the concept of differentiation. We’ve elaborated on its definition and some of its important types which are helpful in the study of calculus, and physics as well as for optimization. In the last section, we solved some examples related to the discussed types. Learn more on how to overcome the fear of mathematics.

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