The differential is one of the main branches of calculus. The other types are limit calculus and integral calculus. These types are frequently used to solve the complex calculus of various kinds. The limit is usually used to define the first two kinds of calculus.
The differential can be taken for a single, double, or multivariable function with respect to variables that are independent. In this post, we will learn all the basics of differential calculus such as definition, types, rules, and solved examples.
What is differential calculus?
A branch of mathematics concerned mostly with the study of the rate of change of a function with respect to its independent variable is said to be differential calculus. In mathematical analysis, the differential is used to concern with the problems of calculus.
The differential can be taken for a constant function, linear function, polynomial function, trigonometric function, exponential function, logarithmic function, etc. All the functions can be differentiated by using the laws of differential calculus.
Kinds of differential calculus
There are various kinds of differential calculus such as

Explicit differential

Implicit differential

Partial differential

Directional differential
Let us briefly describe the above kinds of differential calculus.

Explicit Differential
The commonly used kind of derivative is explicit differential. In this kind of differential calculus, the single variable functions can be differentiated with respect to its independent variable. It is usually denoted by d/dx, d/dy, d/dz, d/du, etc. It is also denoted by f’.

Implicit Differential
The equations are also involved in differentiation. To solve the problems of the equations or implicit functions this kind of differential calculus is frequently used. According to the implicit differential, the notation of differentiation must be applied to both sides of the function.
d/dx f(x, y) = d/dx g(x, y)
Usually, this type of differential finds the derivative of the dependent variable with respect to an independent variable such as dy/dx. It is mainly denoted as y’(x).

Partial Differential
The partial differential is that type of differential that deals with the multivariable functions. The multivariable functions can be differentiated with respect to any variable of the function. e.g., if a function f(x, y, z) is given then the differentiating variable can be x, y, or z.
It is denoted by ∂ f(x, y, z) /∂x, ∂ f(x, y, z) /∂y, or ∂ f(x, y, z) /∂z

Directional Differential
The directional differential is a kind of differentiation that is used to find the direction of the function by using the unit vectors. This kind of differential calculus finds the gradient and normalizes the vectors and takes the dot product of these calculations.
The final result must be the direction of the derivative. This kind of derivative is denoted by ∇u f (x, y).
Laws of differential calculus
Below are some laws of differentiation.
Rules name 
Rules 
Quotient law 
d/du [p(u) / q(u)] = 1/[q(u)]2 [q(u) d/du [p(u)]  p(u) d/du [q(u)]] 
Product law 
d/du [p(u) * q(u)] = q(u) d/du [p(u)]  p(u) d/du [q(u)] 
Constant law 
d/du [k] = 0 
Difference law 
d/du [p(u)  q(u)] = d/du [p(u)  d/du [q(u)] 
Sum law 
d/du [p(u) + q(u)] = d/du [p(u) + d/du [q(u)] 
Exponential law 
d/du [eu] = eu 
Constant function law 
d/du [k * p(u)] = k d/du [p(u)] 
Power law 
d/du [p(u)]n = n [p(u)]n1 * d/du [p(u)] 
How to calculate the differential calculus?
The differential of a function can be calculated easily by using its types and laws. Here are some solved examples of differential calculus.
Example 1
Find the differential of p(u) = 9u2 + 7u4 – 8u3 + 5u + 11 w.r.t “u”.
Solution
Step 1: Write the given differential function and apply the notation of differentiation to it.
p(u) = 9u2 + 7u4 – 8u3 + 5u + 11
d/du p(u) = d/du [9u2 + 7u4 – 8u3 + 5u + 11]
Step 2: Apply the notation of differential calculus to each function separately with the help of the sum and difference laws.
d/du [9u2 + 7u4 – 8u3 + 5u + 11] = d/du [9u2] + d/du [7u4] – d/du [8u3] + d/du [5u] + d/du [11]
Step 3: Take the constant coefficients outside the differential notation.
= 9d/du [u2] + 7d/du [u4] – 8d/du [u3] + 5d/du [u] + d/du [11]
Step 4: Solve the above expression by using the power law.
= 9 [2u21] + 7 [4u41] – 8 [3u31] + 5 [u11] + [0]
= 9 [2u1] + 7 [4u3] – 8 [3u2] + 5 [u0] + [0]
= 9 [2u] + 7 [4u3] – 8 [3u2] + 5 [1] + [0]
= 18u + 28u3 – 24u2 + 5 + [0]
= 18u + 28u3 – 24u2 + 5
The above example of differential calculus can also be calculated with the help of a differentiation calculator to get the stepbystep solution in a couple of seconds.
Example 2:
Differentiate f(u, y) = 5u2y3 + 2u2 – 6y2 + 5y, g(u, y) = (3u5 * 2u3) + 12uy  3y2, w.r.t “u”.
Solution
Step 1: Write the given implicit function and apply the notation of differentiation to it.
5u2y3 + 2u2 – 6y2 + 5y = (3u5 * 2u3) + 12uy  3y2
d/du [5u2y3 + 2u2 – 6y2 + 5y] = d/du [(3u5 * 2u3) + 12uy  3y2]
Step 2: Apply the notation of differential calculus to each function separately with the help of the sum and difference laws.
d/du [5u2y3] + d/du [2u2] – d/du [6y2] + d/du [5y] = d/du [(3u5 * 2u3)] + d/du [12uy]  d/du [3y2]
Step 3: Take the constant coefficients outside the differential notation.
5d/du [u2y3] + 2d/du [u2] – 6d/du [y2] + 5d/du [y] = d/du [(3u5 * 2u3)] + 12d/du [uy]  3d/du [y2]
Step 4: Now apply the product law of differential calculus.
5y3 d/du [u2] + 5u2 d/du [y3] + 2d/du [u2] – 6d/du [y2] + 5d/du [y] = 3u5 d/du [2u3] + 2u3 d/du [3u5] + 12y d/du [u] + 12u d/du [y]  3d/du [y2]
Step 5: Solve the above expression by using the power law.
5y3 [2u21] + 5u2 [3y31 dy/du] + 2 [2u21] – 6 [2y21 dy/du] + 5 [dy/du] = 3u5 [6u31] + 2u3 [15u51] + 12y [u11] + 12u [dy/du]  3 [2y21 dy/du]
5y3 [2u] + 5u2 [3y2 dy/du] + 2 [2u] – 6 [2y dy/du] + 5 [dy/du] = 3u5 [6u2] + 2u3 [15u4] + 12y [1] + 12u [dy/du]  3 [2y dy/du]
10uy3 + 15u2y2 dy/du + 4u – 12y dy/du + 5dy/du = 18u7 + 30u7 + 12y + 12u dy/du  6y dy/du
10uy3 + 15u2y2 dy/du + 4u – 12y dy/du + 5dy/du = 48u7 + 12y + 12u dy/du  6y dy/du
Step 6: Take the dy/du term on a similar side of the equation.
15u2y2 dy/du – 12y dy/du + 5dy/du + 6y dy/du  12u dy/du = 48u7 + 12y  4u  10uy3
(15u2y2 – 12y + 5 + 6y  12u) dy/du = 48u7 + 12y  4u  10uy3
(15u2y2 – 6y + 5  12u) dy/du = 48u7 + 12y  4u  10uy3
dy/du = (48u7 + 12y  4u  10uy3) / (15u2y2 – 6y + 5  12u)
Conclusion
In this article, we have learned all the basics of this kind of calculus. We have thoroughly covered the definition, kinds, and laws of differential calculus along with solved examples. Now you can be a master in differential just by learning this article.
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