Hello, friends in this article you will learn what is derivative of x as well as proof the derivative of x by the power rule and first principal rule, and the last also calculate the numerical problem.
So, without wasting time let's get started.
What are Derivatives
In mathematics, the rate of change of a function with respect to a variable is called a derivative. Derivatives are fundamental to solving problems in calculus and differential equations.
Basic Derivative Rules
The following derivatives rules are used:
- Power Rule
- Product Rule
- Quotient Rule
- Chain Rule
- First Principle Rule
Derivative of x
The derivative of x is always equal to 1. It refers to the result that is derived using different derivatives rules.
There are two main derivatives rules that are used to find the derivative of x. The first one is a power rule and the second one is the first principle rule.
Proof of Derivative of x by Power Rule
The power rule derivatives are the easiest way to evaluate derivatives of x.
The power rule is given by the formula:
d/dx (xⁿ) = nxⁿ⁻¹
Since the exponent of x is 1, to find the derivative of x, by replacing n with 1 in the above formula,
So,
d/dx (x¹) = 1( x¹⁻¹)
d/dx (x) = 1 x⁰
As we know that the value of x⁰ is equal to 1.
So,
The value of the derivative of x will be equal to 1.
Proof of Derivative of x by First Principle
According to the first principle, the derivative limit of a function can be determined by computing the formula:
For a differentiable function y = f (x)
We define its derivative w.r.t x as :
dy/dx = f ' (x) = limₕ→₀ [f(x+h) - f(x)]/h
f'(x) = limₕ→₀ [f(x+h) - f(x)]/h
This limit is used to represent the instantaneous rate of change of the function f(x).
Let
f (x) = x
So,
f(x + h) = x + h
Putting these values on above formula.
So,
f'(x) = limₕ→₀ [(x+h) - x]/h
= limₕ→₀ h/h
= 1
Thus the value of the derivative of x will be equal to 1.
Calculation
Question
What is the derivative of x raised to 6?
Solution
Given,
n = 6
As we know,
The power rule,
d/dx (xⁿ) = nxⁿ⁻¹
d/dx (x⁶) = 6 (x⁶⁻¹)
So putting the value of n then
d/dx (x⁶) = 6 (x⁶⁻¹)
d/dx (x⁶) = 6x⁵
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FAQ Related to Derivative of X
Is the derivative of 1 is x?
No, The rate of change of a function with respect to a variable is called a derivative since 1 is a constant whose value never changes. Thus, the derivative of 1 will be 0.
What is the derivative of -x?
As we know the derivative of x is equal to 1 so - x = -1 therefore the derivative of -x will be equal to the -1.
What is the derivatives of X?
The derivative of x is equal to 1.
How do you find a derivative?
The derivative can be derived by a power rule, product rule, quotient rule, chain rule, first principle rule.
So friends here I discussed all aspects related to the derivative of x.
I hope you enjoy this topic If you have any doubt then you can ask me through comments or direct mail.
Thank You.