Derivative of 2 to the x

Derivative of 2^x is 2^xln2. 2^x Derivative refers to the process of finding change in 2^x function to the independent variable. The specific process of finding the derivative for 2^x functions is referred to as exponential differentiation.

Let’s know more about Derivative of 2^x formula and proof in detail below.

What is Derivative in Math?

The derivative of a function is the rate of change of the function with respect to any independent variable. The derivative of a function f(x) is denoted as f'(x) or (d /dx) [f(x)]. The differentiation of an exponential function is called a derivative of the exponential function or exponential derivatives.

What is Derivative of 2^x?

The derivative of the 2^x is (2^x )·(ln 2). The derivative of 2^x is the rate of change with respect to x.

Derivative of 2^x Formula

The formula for the derivative of 2^x is given by:

d/dx [2^x] = (2^x)·(ln 2)

or

(2^x)’ = (2^x)·(ln 2)

Proof of Derivative of 2^x

The derivative of 2^x can be proved using the following methods:

• By using the First Principle of Derivative
• By using Logarithms

Derivative of 2^x by First Principle of Derivative

To prove derivative of 2^x using First Principle of Derivative, we will use basic limits and exponential formulas which are listed below:

• f'(x) = lim_h→0[f(x + h) – f(x)] / h
• lim_h→0(a^h-1)/h=ln a
• a^m x aⁿ=a^m+n

Let’s start the proof for the derivative of 2^x ,assume that f(x) = 2^x.

By first principle, the derivative of a function f(x) is,

f'(x) = lim_h→0[f(x + h) – f(x)] / h {Using 1}

Since f(x) = 2^x, we have f(x + h) = 2^{(x + h)}.

Substituting these values in (1),

f’ (x) = lim_h→0 [2^{(x + h)} – 2^x]/h

⇒ lim_h→0[2^x · 2^h – 2^x]/h {Using 3}

⇒l im_h→0[2^x · (2^h – 1)]/h

⇒ 2^xlim_h→0[(2^h – 1)]/h {Using 2}

⇒ 2^x · ln 2

Therefore, f'(x) = d/dx [2^x] = 2^x·ln2

Derivative of 2^x by Logarithmic Differentiation

To prove derivative of 2^x using Logarithmic Differentiation, we will use basic formulas which are listed below:

• ln a^b = b ln a
• e^{ln a} = a
• dy/dx[e^x]=e^x

Let’s start the proof for the derivative of 2^x ,assume that y = 2^x.

Taking natural log on both sides,

ln y = ln 2^x

⇒ ln y = x ln 2 {Using 1}

Exponentiate on both sides

⇒ e^{(ln y)} = e^{(x ln 2)}

⇒ y = e^{(x ln 2)} {Using 2}

Take the derivative

⇒ y’ = (ln 2) ·(e^{x ln2}) {Using 3}

⇒ y’ = (ln 2 ) · 2^x

Therefore, f'(x) = d/dx [2^x] = 2^x·ln2

Read More,

• Exponential Functions
• Differentiation Formulas
• Differentiation of Exponential Functions

Solved Examples on Derivative of 2^x

Example 1: Find the derivative of 2^x ·tan x.

Solution:

Let f(x) = 2^x· tan x = u·v

By product rule,

f'(x) = u·v’ + v·u’

⇒ (2^x) d/dx (tan x) + (tan x) d/dx (2^x)

⇒ (2^x)(sec²x) + (tan x) (2^x·ln 2)

Therefore f'(x) = (2^x)(sec2x) + (tan x) (2^x·ln 2)

Example 2: Find the derivative of (2^x)².

Solution:

Let f(x) = (2^x)² = 2^2x

By chain rule,

f'(x) = 2^2x ln 2 d/dx[2x]

⇒ 2·2^2x·ln 2

⇒ 2^2x+1·ln 2

Therefore f'(x)= 2^2x+1·ln 2

Example 3: Find the derivative of 5^x + 5x²

Solution:

Let f(x)=5^x+5x²

f'(x)=5^x·ln 5 + 5·2·x

⇒ 5^x·ln 5 + 10·x

Therefore f'(x)= 5^x·ln 5 + 10·x

Practice Questions Related to Derivative of 2^x

Q1. Find the derivative of 7^x

Q2. Find the derivative of x²·5^x·3^x

Q3. Evaluate: (d/dx) [sec x/(2^x)]

Q4. Evaluate the derivative of: 5^x·5^x·9^x

Q5. Find: (tan x)()

Derivative of 2^x – FAQs

What is Derivative?

A derivative represents the rate at which a function is changing at a particular point. It gives us information about how the function is behaving, whether it’s increasing, decreasing, or staying constant.

Write the Formula for Derivative of 2^x.

The formula for derivative of 2^x is:

(d/dx) 2^x= 2^x· ln 2

What is the Derivative of 2^(-x)?

Derivative of 2^(-x) is 2^(-x)·ln(-x)·(-1)

What are the Different Methods to Prove Derivative of 2^x?

The different methods to prove derivative of 2^x are:

• By using First Principle of Derivative
• By using Logarithms

What is the Derivative of Negative 2^x?

Derivative of negative 2^x i.e., -2^x is (-2^x· ln 2).

What is Derivative of a^x?

Derivative of a^x is a^x·lna.

What is the Derivative of 2·2^x?

Derivative of 2·2^x or 2^x+1 is 2^x+1·ln 2