Derivative of 2 to the x

The derivative of 2 to the x is equal to 2^x ln 2. We can calculate this derivative using various methods of differentiation such as the first principle of derivatives, and the formula for the derivative of the exponential function, and using the natural logarithmic function followed by implicit differentiation. Mathematically, we can write the formula for the derivative of 2 to the x as d(2^x)/dx = 2^x ln 2. The formula for the derivative of the function f(x) = a^x is given by a^x ln a. Using this formula, the derivative of 2 to the x is given by, (2^x)' = 2^x ln 2.

Further, in this article, we will explore the derivative of 2 to the x and its formula using different methods of evaluating derivatives. We will also solve various examples related to the derivative of 2 to the x and other functions for a better understanding of the concept.

The derivative of 2 to the x is 2^x ln 2. We can write this as d/dx (2^x) = 2^x ln 2 (or) (2^x)' = 2^x ln 2. Since "ln" is nothing but natural logarithm (log with base 'e'), we can write this formula as d/dx (2^x) = 2^x logₑ 2. i.e.,

2 to the x is mathematically written as 2^x and it is an exponential function (but NOT a power function). Because its base (2) is a constant and its exponent (x) is a variable. So we use the formula d/dx(a^x) = a^x ln a to find the derivative of 2 to the x but we are not supposed to use the power rule d/dx (xⁿ) = n x^n-1 here as 2^x is NOT a power function.

To prove the derivative of 2 to the x, the straightforward method is using the derivative of exponential function a^x formula which says,

d/dx(a^x) = a^x ln a

Substitute a = 2 on both sides, we get

d/dx(2^x) = 2^x ln 2

Hence the formula is proved.

Derivative of 2 to the x Formula

As observed above, the formula for the derivative of 2 to the x is given by d(2^x)/dx = 2^x ln 2 (or) (2^x)' = 2^x ln 2. There are various other ways to prove the formula of the derivative of 2 to the x. Here are a few of them.

• Using the first principle
• Using logarithmic differentiation
• Using chain rule

Let us prove the formula in each of these cases.

The limit definition of the derivative, which is also known as the first principle, says that the derivative of a function y = f(x) is found by using the limit:

f'(x) = lim_h→0[f(x + h) - f(x)] / h --- (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

Using one of the properties of exponents, a^{m + n} = a^m · aⁿ. Using this, we have

f'(x) = lim_h→0 [2^x · 2^h - 2^x] / h

= lim_h→0 2^x [ 2^h - 1] / h

= lim_h→0 2^x · limₕ→ ₀ [ 2^h - 1] / h

= 2^x · lim_h→0 [ 2^h - 1] / h

Using one of the limit formulas, lim_h→0 [a^h - 1] / h = ln a.

f'(x) = 2^x ln 2

Hence the derivative of 2 to the x formula is proved.

We use logarithmic differentiation to find the derivative of a function that has a variable in the exponent. In this process, we apply "log" (or) "ln" on both sides and then differentiate on both sides. Let us assume the function to be differentiated to be y = 2^x. Taking "ln" on both sides,

ln y = ln 2^x

Using the properties of logarithms, ln a^m = m ln a. Using this,

ln y = x ln 2

Differentiating both sides with respect to x,

d/dx (ln y) = d/dx (x ln 2)

Using the constant multiplication rule of derivatives,

d/dx (ln y) = ln 2 d/dx (x)

Using the derivative of ln x rule, d/dx (ln x) = 1/x and also the chain rule on the left side,

(1/y) dy/dx = ln 2 (1)

Multiplying both sides by y,

dy/dx = y ln 2

Substituting y = 2^x here,

d/dx (2^x) = 2^x ln 2

Hence, we proved the derivative of 2 to the x to be 2^x ln 2. You can try deriving the same formula by applying "log" on both sides.

Using one of the properties of natural logarithms, e^{ln a} = a, for any 'a'. By this, we have

e^{ln 2} = 2 (or) 2 = e^{ln 2}

Raising the exponent both sides by x,

2^x = (e^{ln 2}) ^x

We have (a^m) ⁿ = a^mn. By using this in the above step,

2^x = e^{x ln 2}

Differentiating both sides with respect to x,

d/dx (2^x) = d/dx (e^{x ln 2})

We know that the derivative of e^x is e^x and also applying the chain rule on the right side,

d/dx (2^x) = e^{x ln 2} · d/dx (x ln 2)

= e^{x ln 2} · (ln 2)

= e^{ln 2x} · (ln 2)

Using the same property e^{ln a} = a again,

d/dx (2^x) = 2^x ln 2

Hence, the derivative of 2 to the x formula is derived.

Important Points on Derivative of 2 to the x:

• The derivative of 2 to the x power is, d/dx (2^x) = 2^x ln 2 (or) 2^x logₑ 2.
• Note that 2^x is an exponential function but NOT a power function.
• Use the derivative of a^x formula but NOT the derivative of xⁿ formula to find the derivative of 2 to the x.

☛Related Topics:

• Derivative Rules
• Inverse Trig Derivatives
• Implicit Differentiation

What is the Derivative of 2 to the Power of x?

The derivative of 2 to the power of x has two formulas:

• d/dx (2^x) = 2^x ln 2
• d/dx (2^x) = 2^x logₑ 2

How to Find the Derivative of 2 to the x?

To find the derivative of 2 to the x, just apply the formula d/dx (a^x) = a^x ln a and substitute a = 2 in this formula. Then we get d/dx (2^x) = 2^x ln 2. We can also find the derivative of 2 to the x using the first principle of derivatives, chain rule and implicit differentiation.

Is the Derivative of 2^x Equal to 2^x Itself?

No, the derivative of 2^x is NOT itself, the derivative of 2^x is 2^x ln 2. This comes from the formula d/dx (a^x) = a^x ln a.

What is the n^th Derivative of 2 to the x?

We know that d/dx (2^x) = 2^x ln 2. Let us differentiate it multiple times to identify the pattern.

• The 1^st derivative of 2^x is 2^x ln 2.
• The 2^nd derivative of 2^x is 2^x (ln 2)².
• The 3^rd derivative of 2^x is 2^x (ln 2)³.
• ...
• The n^th derivative of 2^x is 2^x (ln 2)ⁿ.

What is the Derivative of 2 to the x in Terms of Ln?

The derivative of an exponential function is, (a^x) ' = a^x ln a. By substituting a = 2 i this, (2^x)' = 2^x ln 2.

What is the Derivative of 2 to the x in Terms of Log?

The derivative of 2^x is usually expressed in terms of "ln" to be d/dx (2^x) = 2^x ln 2. But we know that ln = logₑ and hence the same formula can be written alternatively to be d/dx (2^x) = 2^x logₑ 2.

What is the Second Derivative of 2 to the x?

The second derivative of 2 to the power x is given by 2^x (ln 2)² and its formula can be written as d²/dx² (2^x) = 2^x (ln 2)².