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Derivative of x
The derivative of x is equal to 1. It refers to the result that is obtained by differentiating x using different methods. Differentiation is the process that is used to find the rate of change of a function. There are two main techniques used when finding the derivative of x. These are the first principle and the differentiation power rule.
In this article, we will see how to find the derivative of x using different methods of evaluating derivatives. We will also solve different examples based on the derivative of functions using the derivative of x for a better understanding of the concept.
|1.||What is the Derivative of x?|
|2.||Derivative of x Formula|
|3.||Derivative of x By First Principle|
|4.||Differentiation of x By Power Rule|
|5.||FAQs on Derivative of x|
What is the Derivative of x?
Differentiation of x is the process of computing the derivative of x. Differentiation is used to denote a small a very small change in a given function with respect to one of its variables. The notation for the differentiation of a function f(x) is given as f'(x) = d[f(x)]/dx. Here, f(x) denotes a function and dx shows the variable with respect to which the function will be differentiated. The differentiation of x can be represented as dx/dx which is equal to 1. We know that the derivative of linear function f(x) = ax + b is equal to a, where a, b are real numbers. For f(x) = x, we have a = 1 and b = 0. Using these facts, we get the derivative of x equal to 1.
Derivative of x Formula
The formula for the derivative of x is given as dx/dx (OR) (x)' = 1. This formula can be evaluated using different methods of differentiation including the first principle of derivatives and power rule of differentiation. The image given below shows the formula for the differentiation of x. Intuitively, the derivative of a function at a point represents the slope of the tangent drawn to the graph of that function at that particular point. As f(x) = x represents a straight line, thus, the derivative of x will be 1 at all points.
Derivative of x By First Principle
The first principle is also known as the definition of a derivative. According to the first principle, the derivative of a function can be determined by calculating the limit formula 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). This formula will be used to evaluate the derivative of x. Let f(x) = x. Thus, f(x + h) = x + h.
dx/dx = f'(x) = limh→0 [x + h - x]/h
= limh→0 h/h
= limh→0 1
Thus, the derivative of x equals 1.
Differentiation of x By Power Rule
The power rule of differentiation is the easiest method to evaluate derivatives of functions of form xn, where n is not equal to -1. The power rule is given as follows: dxn/dx = nxn-1. As the exponent of x is 1 thus, to find the derivative of x, n = 1 needs to be substituted in the aforementioned formula.
dx1/dx = 1 . x1-1
= 1 . x0
= 1 . 1
Thus, using the power rule, the value of the derivative of x is also equal to 1.
Important Notes on Derivative of x
- The derivative of x will be equal to 1. Both the power rule and the first principle can be used to find the derivative of x.
- By using n =1 in the power given by dxn/dx = nxn-1, the derivative of x can be determined.
- As f(x) = x represents a straight line, hence, the derivative will be 1 at all points.
Examples on Derivative of x
Example 1: What is the derivative of x raised to 4 ?
Solution: The derivative of x raised to 4 can be computed using the power rule.
dxn/dx = nxn-1
Here, n = 4
dx4/dx = 4x4-1 = 4x3
Answer: d(x4)/dx = 4x3
Example 2: Find the derivative x raised to 2 using the first principle.
Solution: According to the first principle the formula to compute the derivate is
limh→0 [x + h - x]/h
f(x) = x2 and f(x + h) = (x + h)2
f'(x) = limh→0 [(x + h)2 - x2]/h
Using the algebraic identity (a + b)2 = a2 + 2ab + b2
f'(x) = limh→0 [x2 + h2 + 2xh - x2]/h
f'(x) = limh→0 [h2 + 2xh]/h
f'(x) = limh→0 h [h + 2x]/h
f'(x) = 2x (On applying the limit)
Answer: The derivative of x raised to 2 is 2x.
FAQs on Derivative of x
What is the Derivative of x in Calculus?
The derivative of x in calculus refers to the value that is obtained after the differentiation of x. The power rule and the first principle are used to compute the derivative of x that is equal to 1.
How to Calculate the Derivative of x Using Power Rule?
The power rule of differentiation is given as dxn/dx = nxn-1 = nxn-1. Substituting n = 1 in this formula, the derivative of x can be obtained as 1. We can also evaluate the differentiation of x using the first principle of derivatives.
What is the Formula for Differentiation of x?
The formula for the differentiation of x is dx/dx (OR) (x)' = 1. It can also be given as f'(x) = 1, where f(x) = x.
What is the Derivative of x + 1?
x + 1 can also be written as x1 + x0. By applying the power rule to the first and second terms, the derivative of x + 1 can be computed as 1.
How to Find the nth Derivative of x?
The nth derivative of x can be determined by successive differentiation. The first derivative of x is 1. The second-order derivative of x will be d(1)/dx = 0 because the derivative of a constant function is always zero. Thus, other higher-order derivatives of x will also be 0.
What is the Integral and Derivative of x?
The integral of x is determined by using the power rule of integration as ∫x dx = (x2 / 2) + C, where C is the constant of integration. The derivative of x is equal to 1.
Why is Derivative of x Equal to 1?
The derivative of x is always equal to 1 as it can be proved using the first principle of differentiation. As we evaluate the limit dx/dx = limh→0 [x + h - x]/h, its value is equal to 1. Therefore, the derivative of x is equal to 1.