In mathematics, there are various branches but calculus is the main branch that is widely used to calculate the area under the curve, the numerical value of the function, the slope of the tangent line, etc. This branch of mathematics is conceptual.
There are further two more well-known subtypes of calculus. One is differentiation and the other is integration. Both types are related to each other. In this article, we are going to explain the first subtype of calculus that is differentiation along with examples.
What is differentiation?
Before going to learn how to differentiate the functions, you must have a sound knowledge of the basics of differentiation. Let us describe the term differentiation such as its definition and explanation.
The process of finding the differential of the function with respect to its independent variable is said to be the differentiation calculus. It can also be defined as the instantaneous rate of change of a function with respect to the independent variable of the function.
The term differentiation is also known as the differential, derivative, and differentiate. To study the rate of change of quantities and terms in geometry, differentiation is used. The function can be differentiated with the help of the first principle method or using laws of derivative.
There are 4 well-known types of derivatives in calculus such as:
- Explicit derivative
- Implicit differentiation
- Partial derivative
- Directional derivative
These types of differentiation are used to differentiate the single, double, and multivariable functions with respect to the corresponding variables.
Formula of derivative
There is a general formula of derivative to differentiate the function with the help of the first principle method. Here is the expression of the general formula.
f’(x) = lim_{h}_{→0} f(x + h) – f(x) / h
Laws of derivative
The basic laws of the derivative are:
Laws name |
Laws |
Trigonometry Law |
d/dw [sin(w)] = cos(w) |
Power law |
d/dw [p(w)]^{n} = n [p(w)]^{n-1} * d/dw [p(w)] |
Quotient law |
d/dw [p(w) / q(w)] = 1/[q(w)]^{2} [q(w) d/dw [p(w)] – p(w) d/dw [q(w)]] |
Product law |
d/dw [p(w) * q(w)] = q(w) d/dv [p(w)] – p(w) d/dw [q(w)] |
Constant law |
d/dw [k] = 0 |
Difference law |
d/dw [p(w) – q(w)] = d/dw [p(w)] – d/dw [q(w)] |
Sum law |
d/dw [p(w) + q(w)] = d/dw [p(w)] + d/dw [q(w)] |
Exponential law |
d/dw [e^{w}] = e^{w} |
Constant function law |
d/dv [k * p(v)] = k d/dv [p(v)] |
How to differentiate the functions?
Follow the laws of the derivative or first principle method to differentiate the functions. You can differentiate the functions easily either by using a differential calculator or manually. Let us describe how to differentiate the function by using a calculator and manually.
1. By using a derivative calculator
There are online tools available to calculate the problems with steps in a fraction of a second. These tools are helpful in solving any kind of numerical problem. A derivative calculator is a helpful tool to differentiate the functions without doing time-consuming calculations.
How to use this calculator?
Follow the below steps.
- Enter the function into the required input field.
- Select the corresponding variable “x” is selected by default.
- Enter the number of derivatives you want to calculate i.e., 1 for the first derivative, 2 for the second, and so on.
- Press the calculate button.
- The solution with steps will come below the calculate button.
Manually
Here are a few examples to differentiate the functions manually.
Example 1: By the first principle method
Differentiate the given function with respect to “w”.
f(w) = w + 4
Solution
Step 1: Write the formula of differentiation.
d/dw [f(w)] = lim_{h}_{→0} f(w + h) – f(w) / h
Step 2: Now substitute the function in the formula by making the terms.
f(w) = w + 4
f(w + h) = w + h + 4
d/dw [w + 4] = lim_{h}_{→0} [w + h + 4 – (w + 4)] / h
d/dw [w + 4] = lim_{h}_{→0} [w + h + 4 – w – 4] / h
d/dw [w + 4] = lim_{h}_{→0} [h] / h
d/dw [w + 4] = lim_{h}_{→0} [1]
Step 3: Now apply the limit
d/dw [w + 4] = lim_{h}_{→0} [1] = 1 (limit of a constant function remain the same)
Example 2: By using the laws
Differentiate the given function with respect to “z”.
f(z) = 3z^{4} + 4z^{3} – 2z^{2} + 5z + 320
Solution
Step 1: First of all, write the given information of the function and apply the notation of differentiation to the function.
f(z) = 3z^{4} + 4z^{3} – 2z^{2} + 5z + 320
corresponding variable = z
d/dz f(z) = d/dz [3z^{4} + 4z^{3} – 2z^{2} + 5z + 320]
Step 2: Apply the notation of differential to each function separately with the help of difference and sum laws.
d/dz [3z^{4} + 4z^{3} – 2z^{2} + 5z + 320] = d/dz [3z^{4}] + d/dz [4z^{3}] – d/dz [2z^{2}] + d/dz [5z] + d/dz [320]
Step 3: Now use the constant function law of derivative to take the constant coefficients outside the notation.
d/dz [3z^{4} + 4z^{3} – 2z^{2} + 5z + 320] = 3d/dz [z^{4}] + 4d/dz [z^{3}] – 2d/dz [z^{2}] + 5d/dz [z] + d/dz [320]
Step-4: Now differentiate the above expression with the help of the power rule.
d/dz [3z^{4} + 4z^{3} – 2z^{2} + 5z + 320] = 3 [4 z^{4-1}] + 4 [3 z^{3-1}] – 2 [2 z^{2-1}] + 5 [z^{1-1}] + [0]
d/dz [3z^{4} + 4z^{3} – 2z^{2} + 5z + 320] = 3 [4 z^{3}] + 4 [3 z^{2}] – 2 [2 z^{1}] + 5 [z^{0}]
d/dz [3z^{4} + 4z^{3} – 2z^{2} + 5z + 320] = 3 [4 z^{3}] + 4 [3 z^{2}] – 2 [2 z] + 5 [1]
d/dz [3z^{4} + 4z^{3} – 2z^{2} + 5z + 320] = 12 [z^{3}] + 12 [z^{2}] – 4 [z] + 5 [1]
d/dz [3z^{4} + 4z^{3} – 2z^{2} + 5z + 320] = 12z^{3} + 12z^{2} – 4z + 5
Example 3
Differentiate the given function with respect to “θ”.
f(θ) = 12θ^{4} + 16θ^{3} – 2θ^{2} + 5θ + 20
Solution
Step 1: First of all, write the given information of the function and apply the notation of differentiation to the function.
f(θ) = 12θ^{4} + 16θ^{3} – 2θ^{2} + 5θ + 20
corresponding variable = θ
d/dθ f(θ) = d/dθ [12θ^{4} + 16θ^{3} – 2θ^{2} + 5θ + 20]
Step 2: Apply the notation of differential to each function separately with the help of difference and sum laws.
d/dθ [12θ^{4} + 16θ^{3} – 2θ^{2} + 5θ + 20] = d/dθ [12θ^{4}] + d/dθ [16θ^{3}] – d/dθ [2θ^{2}] + d/dθ [5θ] + d/dθ [20]
Step 3: Now use the constant function law of derivative to take the constant coefficients outside the notation.
d/dθ [12θ^{4} + 16θ^{3} – 2θ^{2} + 5θ + 20] = 12d/dθ [θ^{4}] + 16d/dθ [θ^{3}] – 2d/dθ [θ^{2}] + 5d/dθ [θ] + d/dθ [20]
Step 4: Now differentiate the above expression with the help of the power rule.
d/dθ [12θ^{4} + 16θ^{3} – 2θ^{2} + 5θ + 20] = 12 [4 θ^{4-1}] + 16 [3 θ^{3-1}] – 2 [2 θ^{2-1}] + 5 [θ^{1-1}] + [0]
d/dθ [12θ^{4} + 16θ^{3} – 2θ^{2} + 5θ + 20] = 12 [4 θ^{3}] + 16 [3 θ^{2}] – 2 [2 θ^{1}] + 5 [θ^{0}]
d/dθ [12θ^{4} + 16θ^{3} – 2θ^{2} + 5θ + 20] = 12 [4 θ^{3}] + 16 [3 θ^{2}] – 2 [2 θ] + 5 [1]
d/dθ [12θ^{4} + 16θ^{3} – 2θ^{2} + 5θ + 20] = 48 [θ^{3}] + 48 [θ^{2}] – 4 [θ] + 5 [1]
d/dθ [12θ^{4} + 16θ^{3} – 2θ^{2} + 5θ + 20] = 48θ^{3} + 48θ^{2} – 4θ + 5
Final words
Now you can differentiate any function with respect to any variable just by following the above post. In this article, we have discussed all the basics of how to differentiate the function either by using laws of differentiation or the first Principe method.
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