Derivative of 2x
The derivative of 2x is equal to 2 as the formula for the derivative of a straight line function f(x) = ax + b is given by f'(x) = a, where a, b are real numbers. Differentiation of 2x is calculated using the formula d(ax+b)/dx = a. We can also evaluate the derivative of 2x using the differentiation power rule which has the formula d(x^{n})/dx = nx^{n1}. Derivative of 2x can also be determined using other methods of differentiation.
Further in this article, we will evaluate the derivative of 2x using various differentiation methods and its formula. We will prove the differentiation of 2x and go through some solved examples involving the derivative of 2x for a better understanding of the concept.
1.  What is the Derivative of 2x? 
2.  Derivative of 2x Formula 
3.  Derivative of 2x Proof 
4.  FAQs on Derivative of 2x 
What is the Derivative of 2x?
The derivative of a function gives the rate of change in that function with respect to the change in the variable. For a linear function f(x) = ax + b, the derivative is a constant function. Hence, the derivative of 2x is a constant which is given by 2. We can evaluate the derivative of 2x using different methods of differentiation such as the power rule, product rule, first principle of derivatives, and the derivative of linear function formula. Also, as we know that the derivative of kx is equal to k, therefore it follows that the derivative of 2x is 2. Let us now see the formula for the differentiation of 2x.
Derivative of 2x Formula
The formula for the derivative of 2x is given by d(2x)/dx = 2. We can compute the differentiation of 2x using the fact that the derivative of f(x) = kx is equal to f'(x) = k. Using this, we can say that the derivative of 2x is equal to 2. The image below shows the formula for the differentiation of 2x:
Derivative of 2x Proof
Now that we know that the derivative of 2x is equal to 2, we will derive this using various rules of derivatives. We can derive the formula using the definition of derivatives using limits, power rule, product rule, and the derivative of f(x) = ax + b formula.
Derivative of 2x Using Limits
To derive the derivative of 2x using the first principle of derivatives, we will use the following formulas:
 d(f(x))/dx = lim_{h→0} [f(x+h)  f(x)]/h
 lim_{h→0} k = k, where k is a constant
d(2x)/dx = lim_{h→0} [2(x+h)  2x]/h
= lim_{h→0} [2x + 2h  2x]/h
= lim_{h→0} [2h]/h
= lim_{h→0} 2
= 2
Hence, the derivative of 2x is equal to 2 using the first principle of derivatives.
Differentiation of 2x Using Power Rule
The power rule of differentiation states that the derivative of x to the power n is given by n times x to the power n minus 1, that is, d(x^{n})/dx = n x^{n1}. We will also use the derivative rule of scalar multiple of a function, that is, d(kf(x))/dx = kd(f(x))/dx. Therefore, we have d(2x)/dx = 2 dx/dx = 2. Hence, the differentiation of 2x is equal to 2.
Derivative of 2x Using Product Rule
The product rule of differentiation is used to find the derivative of the product of two or more functions. If we have h(x) = f(x)g(x), then the derivative of h(x) is given by, h'(x) = f'(x) g(x) + f(x) g'(x). Similarly, for h(x) = 2x, we have f(x) = 2 and g(x) = x. Using the product rule, we have,
(2x)' = (2)' × x + 2 × (x)'
= 0 × x + 2 × 1
= 2
Therefore, we have that the derivative of 2x is 2 using the product rule.
Important Notes on Derivative of 2x
 The derivative of 2x is 2 which can be derived using different methods of differentiation.
 We can use the power rule, product rule, and first principle of derivatives, we can derive the differentiation of 2x.
 Using the formula, [kx]' = k, we have that the derivative of 2x is given by, [2x]' = 2.
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Derivative of 2x Examples

Example 1: Evaluate the derivative of 2x / (1 + x^{2}).
Solution: To determine the derivative of 2x / (1 + x^{2}), we will use the derivative of 2x and the quotient rule of derivatives which is d(f/g)/dx = [f'g  fg']/g^{2}. Here f(x) = 2x, g(x) = 1 + x^{2}. Therefore, the derivative is given by,
[2x / (1 + x^{2})]' = [(2x)' (1 + x^{2})  2x (1 + x^{2})']/(1 + x^{2})^{2}
= [2 × (1 + x^{2})  2x × 2x]/(1 + x^{2})^{2}
= [2(1 + x^{2}  2x^{2})]/(1 + x^{2})^{2}
= [2 (1  x^{2})]/(1 + x^{2})^{2}
Answer: Therefore, the derivative of 2x / (1 + x^{2}) is [2 (1  x^{2})]/(1 + x^{2})^{2}

Example 2: Determine the derivative of 2x + 1. Check if it is equal to the derivative of 2x.
Solution: To find the derivative of 2x + 1, we will use the formula d(ax + b)/dx = a.
As we know that for f(x) = 2x, we have a = 2, b = 0, and for g(x) = 2x + 1, we have a = 2, b = 1.
This implies that the derivative of 2x + 1 is (2x + 1)' = 2 and the derivative of 2x is also 2.
Answer: Derivative of 2x + 1 is 2 and (2x + 1)' = (2x)' = 2
FAQs on Derivative of 2x
What is the Derivative of 2x?
The derivative of 2x is equal to 2 as the derivative of the function f(x) = kx is given by f'(x) = k.
What is the Formula for the Differentiation of 2x?
The formula for the differentiation of 2x is given by (2x)' = 2 which is a constant function as the derivative of a linear polynomial f(x) = ax + b is constant f'(x) = a.
How To Find The Derivative of 2x?
We can find the derivative of 2x using different rules of differentiation such as the power rule, scalar multiple of a function, and the first principle of derivatives.
What is the Derivative of 2x / (1  x^{2})?
The derivative of 2x / (1  x^{2}) is given by 2(1 + x^{2})/(1  x^{2})^{2}. This derivative can be evaluated using the quotient rule of differentiation.
What is the Second Derivative of 2x?
The second derivative of 2x can be determined by differentiating the first derivative of 2x. The first derivative of 2x is equal to 2 which is a constant function and the derivative of a constant function is zero. Therefore, the second derivative of 2x is equal to 0.
Which Formula Can be Used to Find the Derivative of 2x?
We can use different formulas of differentiation to find the derivative 2x such as:
 First principle of derivatives: d(f(x))/dx = lim_{h→0} [f(x+h)  f(x)]/h
 Product Rule: h'(x) = f'(x) g(x) + f(x) g'(x), where h(x) = f(x) g(x)
 Derivative of kx: [kx]' = k
What is the Derivative of 2x Square?
The derivative of 2x square, that is, 2x^{2} is determined using the power rule of derivatives. We have (2x^{2})' = 4x. Therefore, the derivative of 2x^{2} is equal to 4x.
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