What is differentiation? Explained with examples
As it is the era of science, mathematics is an essential and useful subject that is frequently used in solving numerical problems and theorems. There are further kinds of mathematics that are helpful in solving daily life problems.
One of the main branches of mathematics is calculus. It is a general method of calculations and reasoning. Differential calculus and integral calculus are two commonly used types of calculus. In this post, we’ll learn the basics of differentiation along with calculations.
Differentiation – Definition
In calculus, differentiation is the process of finding the derivative of the function with respect to the independent variable. In general, the rate of change of a function with respect to an independent variable is known as a derivative.
It is also known as differential calculus. The differential is widely used in geometry in which the rate of change of quantities is studied. The laws of differentiation are very helpful in finding the derivative of the functions.
The single variable, double variable, and multivariable function can be differentiated according to the variables. It can also be differentiated by using the first principle method. The general expression of this method is:
f’(x) = lim_{h}_{→0} f(x + h) – f(x) / h
Laws of differentiation
Here are some laws of differentiation.
Laws name |
Laws |
Quotient law |
d/dv [p(v) / q(v)] = 1/[q(v)]^{2} [q(v) d/dv [p(v)] – p(v) d/dv [q(v)]] |
Product law |
d/dx [a(x) * b(x)] = q(v) d/dv [p(v)] – p(v) d/dv [q(v)] |
Constant law |
d/dv [k] = 0 |
Difference law |
d/dv [p(v) – q(v)] = d/dv [p(v)] – d/dv [q(v)] |
Sum law |
d/dv [p(v) + q(v)] = d/dv [p(v)] + d/dv [q(v)] |
Exponential law |
d/dv [e^{v}] = e^{v} |
Constant function law |
d/dv [k * p(v)] = k d/dv [p(v)] |
Power law |
d/dv [p(v)]^{n} = n [p(v)]^{n-1} * d/dv [p(v)] |
How to calculate the differentiation problems?
The differentiation problems can be solved by using two methods.
- By using a differentiation calculator
- Manually
Let us briefly describe both methods.
By using a differentiation calculator
There are hundreds of tools present online for solving numerical problems. The online tools provide solutions with steps to complex problems. A differentiation calculator is a helpful tool that can solve complex differential problems with steps with a single click.
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
- The solution with steps will come below the calculate button.
Manually
To solve the problems of differentiation manually follow the below steps.
Example-1
Calculate the differentiation of the given function w.r.t “v”.
p(v) = 12v^{5} + 3v^{4} – 24v^{2} + 54v + 120
Solution
Step-1: Take the given function and apply the differential notation to it.
p(v) = 12v^{5} + 3v^{4} – 24v^{2} + 54v + 120
Independent variable = v
d/dv p(v) = d/dv [12v^{5} + 3v^{4} – 24v^{2} + 54v + 120]
Step-2: Now separate each function with the help of sum & difference laws and apply the differentiation notation.
d/dv [12v^{5} + 3v^{4} – 24v^{2} + 54v + 120] = d/dv [12v^{5}] + d/dv [3v^{4}] – d/dv [24v^{2}] + d/dv [54v] + d/dv [120]
Step-3: Take the constant coefficients outside the derivative notation with the help of the constant function law of the differential notation.
d/dv [12v^{5} + 3v^{4} – 24v^{2} + 54v + 120] = 12 d/dv [v^{5}] + 3d/dv [v^{4}] – 24d/dv [v^{2}] + 54d/dv [v] + d/dv [120]
Step-4: Now differentiate the above expression with the help of the power rule.
d/dv [12v^{5} + 3v^{4} – 24v^{2} + 54v + 120] = 12 [5v^{5-1}] + 3 [4v^{4-1}] – 24 [2v^{2-1}] + 54 [v^{1-1}] + [0]
d/dv [12v^{5} + 3v^{4} – 24v^{2} + 54v + 120] = 12 [5v^{4}] + 3 [4v^{3}] – 24 [2v^{1}] + 54 [v^{0}] + [0]
d/dv [12v^{5} + 3v^{4} – 24v^{2} + 54v + 120] = 12 [5v^{4}] + 3 [4v^{3}] – 24 [2v] + 54 [1] + [0]
d/dv [12v^{5} + 3v^{4} – 24v^{2} + 54v + 120] = 12 * 5v^{4} + 3 * 4v^{3} – 24 * 2v + 54 [1]
d/dv [12v^{5} + 3v^{4} – 24v^{2} + 54v + 120] = 60v^{4} + 12v^{3} – 48v + 54
Example 2
Calculate the differentiation of the given function w.r.t “r”.
p(r) = 12r^{5} + 2r^{3} – 7sin(r) + 5r + 12cos(r)
Solution
Step-1: Take the given function, and independent variable, and apply the notation of differentiation to it.
p(r) = 12r^{5} + 2r^{3} – 7sin(r) + 5r + 12cos(r)
Independent variable = r
d/dr p(r) = d/dr [12r^{5} + 2r^{3} – 7sin(r) + 5r + 12cos(r)]
Step-2: Now separate each function with the help of sum and difference laws and apply the notation of differential calculus.
d/dr [12r^{5} + 2r^{3} – 7sin(r) + 5r + 12cos(r)] = d/dr [12r^{5}] + d/dr [2r^{3}] – d/dr [7sin(r)] + d/dr [5r] + d/dr [12cos(r)]
Step-3: Take the constant coefficients outside the derivative notation with the help of the constant function law of the differential notation.
d/dr [12r^{5} + 2r^{3} – 7sin(r) + 5r + 12cos(r)] = 12d/dr [r^{5}] + 2d/dr [r^{3}] – 7d/dr [sin(r)] + 5d/dr [r] + 12d/dr [cos(r)]
Step-4: Now differentiate the above expression with the help of power and trigonometry laws.
d/dr [12r^{5} + 2r^{3} – 7sin(r) + 5r + 12cos(r)] = 12 [5r^{5-1}] + 2 [3r^{3-1}] – 7 [cos(r)] + 5 [r^{1-1}] + 12 [-sin(r)]
d/dr [12r^{5} + 2r^{3} – 7sin(r) + 5r + 12cos(r)] = 12 [5r^{4}] + 2 [3r^{2}] – 7 [cos(r)] + 5 [r^{0}] + 12 [-sin(r)]
d/dr [12r^{5} + 2r^{3} – 7sin(r) + 5r + 12cos(r)] = 12 [5r^{4}] + 2 [3r^{2}] – 7 [cos(r)] + 5 [1] + 12 [-sin(r)]
d/dr [12r^{5} + 2r^{3} – 7sin(r) + 5r + 12cos(r)] = 12 * 5r^{4} + 2 * 3r^{2} – 7 [cos(r)] + 5 [1] + 12 [-sin(r)]
d/dr [12r^{5} + 2r^{3} – 7sin(r) + 5r + 12cos(r)] = 60r^{4} + 6r^{2} – 7cos(r) + 5 – 12sin(r)
Wrap up
In this post, we have covered all the basic intent of the differentiation along with its definition, formula, laws, and solved examples. Now you can solve any differentiation problem either by using the above-mentioned calculator or a manual method.