In mathematics with the help of the rules of derivatives, you can calculate the cost, amount of the material used in an apartment, strength, loss, and many more such things, and also to calculate the maximum and minimum values of functions, we use derivatives to do such things. A derivative is usually a type of calculus.
Derivatives are widely used in mathematics, engineering, and different branches of science to calculate the problems related to finding the change in the function concerning the independent variable.
Table of Contents
What are derivatives?
In calculus, the derivative is usually defined as the rate of change of the function with respect to the independent variable. Derivatives are generally represented by the slope of the graph or the slope having a tangent line at a particular point.
Derivatives are also stated as the slope of the line that lies at a particular point to the tangent of the curve. For the calculation of derivatives, limits play an important role in this regard. As we can calculate derivatives by using limits this process is known as the first principle.
Derivative with respect to x by first principal,
d/dx (f(x)) = limh→0 f(x + h) – f(x) / h
- To reverse this process of taking derivative, we use integration also known as the antiderivative.
What are the different rules of Derivatives?
There are various rules of derivatives. Let’s discuss them briefly.
Rule of a constant function
In general, the derivative of a constant function or a constant term is always zero. Constants are all those functions that do not have any term of the variable whose you want to calculate the derivative. Such as if you want to calculate the derivative with respect to x then each term that does not have x will consider as constant.
d/dx (A) = 0
Example
Calculate the derivative of 4 with respect to x.
Solution
Step 1: Identify the function.
F(x) = 4
Step 2: Take the formula of derivative.
d/dx (f(x))
Step 3: Put the value of the function.
d/dx (4)
Step 4: Apply constant function rule.
d/dx (4) = 0
Rule of a Power function
If the function is given in power, then this rule is applicable. In this rule, power must be multiplied by the coefficient if there is no coefficient then multiply the power by one and subtract one from the exponent.
d/dx xn = nxn-1
Example
Calculate the derivative of x4 with respect to x.
Solution
Step 1: Identify the function.
F(x) = x4
Step 2: Take the formula of derivative.
d/dx (f(x))
Step 3: Put the value of the function.
d/dx (x4)
Step 4: Apply power of the function rule.
d/dx (x4) = 4x4-1 = 4x3
Rule of the sum of the functions
When two functions are given having an addition sign among them, then this rule is applicable. In this rule, the derivative is applied to both functions separately.
d/dx (f(x) + g(x)) = d/dx (f(x)) + d/dx (g(x))
Example
Calculate the derivative of 4x + 3x2 with respect to x.
Solution
Step 1: Identify the function.
F(x) = 4x + 3x2
Step 2: Take the formula of derivative.
d/dx (f(x))
Step 3: Put the value of the function.
d/dx (4x + 3x2)
Step 4: Apply sum of the function rule.
d/dx (4x + 3x2) = d/dx (4x) + d/dx (3x2)
= 4x1-1 + (3 x 2) x2-1
= 4x0 + (6)x1
= 4(1) + 6x
d/dx (4x + 3x2) = 4 + 6x
Rule of the Difference of the functions
When two functions are given having a subtraction sign among them, then this rule is applicable. In this rule, the derivative is applied to both functions separately.
d/dx (f(x) – g(x)) = d/dx (f(x)) – d/dx (g(x))
Example
Calculate the derivative of 4x – 3x2 with respect to x.
Solution
Step 1: Identify the function.
F(x) = 4x – 3x2
Step 2: Take the formula of derivative.
d/dx (f(x))
Step 3: Put the value of the function.
d/dx (4x – 3x2)
Step 4: Apply difference of the function rule.
d/dx (4x – 3x2) = d/dx (4x) – d/dx (3x2)
= 4x1-1 – (3 x 2) x2-1
= 4x0 – (6)x1
= 4(1) – 6x
d/dx (4x – 3x2) = 4 – 6x
Rule of the Product of the functions
When two functions are given having a multiplication sign among them, then this rule is applicable. In this rule, the derivative is applied to both functions by an additional method.
d/dx (f(x) * g(x)) = f(x) d/dx (g(x)) + g(x) d/dx (f(x))
Example
Calculate the derivative of 4x * 3x2 with respect to x.
Solution
Step 1: Identify the function.
F(x) = 4x * 3x2
Step 2: Take the formula of derivative.
d/dx (f(x))
Step 3: Put the value of the function.
d/dx (4x * 3x2)
Step 4: Apply product of the function rule.
d/dx (4x * 3x2) = 4x d/dx (3x2) + 3x2 d/dx (4x)
= 4x (3×2 x2-1) + 3x2 (4x1-1)
= 4x (6x) + 3x 2 (4)
= 24x2 + 12x2
= 36x 2
Rule of the Quotient of the functions
When two functions are given having a quotient sign among them, then this rule is applicable. In this rule, the derivative is applied to both functions separately.
d/dx (f(x) / g(x)) = 1/g(x)2[g(x) d/dx (f(x)) – f(x) d/dx (g(x))]
Example
Calculate the derivative of 4x / 3x2 with respect to x.
Solution
Step 1: Identify the function.
F(x) = 4x / 3x2
Step 2: Take the formula of derivative.
d/dx (f(x))
Step 3: Put the value of the function.
d/dx (4x / 3x2)
Step 4: Apply quotient of the function rule.
d/dx (4x / 3x2) = 1/(3x2)2[3x2 d/dx(4x) – 4x d/dx(3x2)]
= 1/9x4[3x2 (4x1-1) – 4x (3×2 x2-1)]
= 1/9 x 4 [3x 2 (4) – 4x (6x)]
= 1/9 x 4 [12 x 2 – 24 x 2 ]
= 1/9x4(-12x2)
= -4/3x 2
How to calculate Derivatives?
To calculate the derivatives of the functions, keep all the rules in your mind for the calculation of the functions. If you want to do these problems accurately and, in few seconds, you can use derivative calculator which gives the result with steps.
Example
Calculate the derivative of (4x + 3) * (3x – 4x2) with respect to x.
Solution
Step 1: Identify the function.
F(x) = (4x + 3) * (3x – 4x2)
Step 2: Take the formula of derivative.
d/dx (f(x))
Step 3: Put the value of the function.
d/dx ((4x + 3) * (3x – 4x2))
Step 4: Apply product of the function rule.
d/dx ((4x + 3) * (3x – 4x2)) = (4x + 3) d/dx (3x – 4x2) + (3x – 4x2) d/dx (4x +3)
Step 5: Apply sum, difference, power, and constant rules.
d/dx ((4x + 3) * (3x – 4x2)) = (4x + 3) d/dx (3x – 4x2) + (3x – 4x2) d/dx (4x +3)
= (4x + 3) (d/dx (3x) – d/dx (4x2)) + (3x – 4x2) (d/dx (4x) + d/dx (3))
= (4x + 3) ((3x1-1) – (4×2 x2-1)) + (3x – 4x2) ((4x1-1) + 0)
= (4x + 3) (3 – 8x) + (3x – 4x 2 ) (4 + 0)
= (4x + 3) (3 – 8x) + (3x – 4x 2 ) (4)
= 12x – 32x2 + 9 – 24x + 12x – 16x2
= 24x – 48x2 – 24x + 9
= -48x2 + 9
Summary
A derivative is not much difficult topic. Once you grab the basic knowledge of the rule of the derivatives you can easily solve all the problems related to finding the derivatives of the functions.
