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## Differential calculus: Its definition, rules, and how to calculate it?

Calculus is a sub-branch of mathematics that is used to define, explain, and calculate the area under the curve, the slope of the tangent line, etc. Calculus is usually used to deal with differentiation and integration.

The term differentiation is relevant to determining the slope of the tangent line while integration is helpful in determining the area under the curve. In this lesson, we will discuss the definition and rules of differentiation and how to evaluate it along with calculations.

## What is differential calculus?

The term differential calculus in mathematics is the study of the rate of change of quantities according to their respective variables. It is one of the traditional divisions of calculus, along with integral calculus. Integral calculus studies the area under a curve.

Differential calculus is also known as derivative or differentiation. The process of finding the differential of the function with respect to the corresponding variable is said to be the differentiation. It is the instantaneous rate of change of the function according to the independent variable.

It is denoted by d/dz or f’(z). The rules of differentiation and the first principle method in which limits are involved are frequently used to evaluate the differential of complex calculus problems.

Let us take the general rules and first principle equation that are used to evaluate the differential of the functions.

### Equation of first principle method

Here is the equation of the first principle method that is helpful in calculating the derivative of the function.

d/dz [f(z)] = limh→0 [f(z + h) – f(z)] / h

### Rules of differentiation

Here are the basic and well-known rules of differentiation that are used to evaluate complex calculus problems to determine the differential of the given functions.

 Rules name Rules Trigonometry Rule d/dz [sin(z)] = cos(z)d/dz [cos(z)] = -sin(z)d/dz [tan(z)] = sec2(z) Power Rule d/dz [f(z)]n = n [f(z)]n-1 * [f’(z)] Quotient Rule d/dz [f(z) / g(z)] = 1/[g(z)]2 [g(z) * [f’(z)] - f(z) * [g’(z)]] Product Rule d/dz [f(z) * g(z)] = g(z) * [f’(z)] + f(z) * [g’(z)] Constant Rule d/dz [L] = 0, where L is any constant Difference Rule d/dz [f(z) - g(z)] = [f’(z)] - [g’(z)] Sum Rule d/dz [f(z) + g(z)] = [f’(z)] + [g’(z)] Exponential Rule d/dz [ez] = ez d/dz [z] Constant function Rule d/dz [L * f(z)] = L [f’(z)]

You can get help through a differential calculator to differentiate complex calculus problems according to the rules of differentiation.

## How to calculate differential calculus?

There are two ways two calculate the differential of the function one is by using the first principle method in which limit calculus is used while the is by using the rules of differentiation. Here are a few examples of differential calculus to learn how to calculate it.

Example 1: By the first principle method

Calculate the differential of the given function with respect to “z”.

f(z) = 12z2 – 3z - 12

Solution

Step 1: First of all, take the general equation of the first principle method that is helpful in calculating the derivative.

d/dz [f(z)] = limh→0 [f(z + h) – f(z)] / h

Step 2: Now take the given function f(z) and determine the f(z + h).

f(z) = 12z2 – 3z – 12

f(z + h) = 12(z + h)2 – 3(z + h) – 12

f(z + h) = 12(z2 + zh + h2) – 3(z + h) – 12

f(z + h) = 12z2 + 12zh + 12h2 – 3z – 3h  – 12

Step 3: Now substitute the values in the general expression of the first principle method

d/dz [12z2 – 3z – 12] = limh→0 [12z2 + 12zh + 12h2 – 3z – 3h – 12] – [12z2 – 3z – 12] / h

d/dz [12z2 – 3z – 12] = limh→0 [12z2 + 12zh + 12h2 – 3z – 3h – 12 – 12z2 + 3z + 12] / h

d/dz [12z2 – 3z – 12] = limh→0 [12z2 + 12zh + 12h2 – 3z – 3h – 12 – 12z2 + 3z + 12] / h

d/dz [12z2 – 3z – 12] = limh→0 [12zh + 12h2 – 3h] / h

d/dz [12z2 – 3z – 12] = limh→0 [h(12z + 12h – 3)] / h

d/dz [12z2 – 3z – 12] = limh→0 [12z + 12h – 3]

d/dz [12z2 – 3z – 12] = limh→0 [12z] + limh→0 [12h] – limh→0 [3]

d/dz [12z2 – 3z – 12] = 12z + 12(0) – 3

d/dz [12z2 – 3z – 12] = 12z – 3

Example 2: By using the laws

Calculate the differential of the given function with respect to “w”.

f(w) = 3w2 – 5w + 7w3 + 6w2 * w3

Solution

Step 1: First of all, write the given function f(w) and the independent variable of the function and apply the notation of differential to the function.

f(w) = 3w2 – 5w + 7w3 + 6w2 * w3

corresponding variable = w

d/dw f(w) = d/dw [3w2 – 5w + 7w3 + 6w2 * w3]

Step 2: Now take the given function and apply the rules of differentiation to it and apply the notation of differential separately to it.

d/dw [3w2 – 5w + 7w3 + 6w2 * w3] = d/dw [3w2] – d/dw [5w] + d/dw [7w3] + d/dw [6w2 * w3]

Step 3: Now apply the product rule of differential calculus.

d/dw [3w2 – 5w + 7w3 + 6w2 * w3] = d/dw [3w2] – d/dw [5w] + d/dw [7w3] + [d/dw [6w2] * w3 + 6w2 * d/dw[w3]]

d/dw [3w2 – 5w + 7w3 + 6w2 * w3] = d/dw [3w2] – d/dw [5w] + d/dw [7w3] + w3 d/dw [6w2] + 6w2 d/dw[w3]

Step 4: Now take the constant coefficients outside the differential notation.

d/dw [3w2 – 5w + 7w3 + 6w2 * w3] = 3d/dw [w2] – 5d/dw [w] + 7d/dw [w3] + 6w3 d/dw [w2] + 6w2 d/dw[w3]

Step 5: Now differentiate the above expression with the help of the power rule.

d/dw [3w2 – 5w + 7w3 + 6w2 * w3] = 3 [2 w2-1] – 5 [w1-1] + 7 [3 w3-1] + 6w3 [2 w2-1] + 6w2 [3 w3-1]

d/dw [3w2 – 5w + 7w3 + 6w2 * w3] = 3 [2 w1] – 5 [w0] + 7 [3 w2] + 6w3 [2 w1] + 6w2 [3 w2]

d/dw [3w2 – 5w + 7w3 + 6w2 * w3] = 3 [2 w] – 5 [1] + 7 [3 w2] + 6w3 [2 w] + 6w2 [3 w2]

d/dw [3w2 – 5w + 7w3 + 6w2 * w3] = 6w – 5 + 21w2 + 12w4 + 18w4

d/dw [3w2 – 5w + 7w3 + 6w2 * w3] = 6w – 5 + 21w2 + 30w4

d/dw [3w2 – 5w + 7w3 + 6w2 * w3] = 6w – 5 + 21w2 + 30w4

d/dw [3w2 – 5w + 7w3 + 6w2 * w3] = 30w4 + 21w2 + 6w – 5

## Conclusion

Now you can get all the basics of differential calculus and how to calculate it from this post. We have discussed each concept of this topic by taking care of the intent along with solved examples.

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