Derivative of Logarithmic Functions in Calculus

Derivative or Differentiation of Logarithmic Function as the name suggests, explores the derivatives of log functions with respect to some variable. As we know, derivatives are the backbone of Calculus and help us solve various real-life problems. Derivatives of the log functions are used to solve various differentiation of complex functions involving logarithms. The differentiation of logarithmic functions makes the product, division, and exponential complex functions easier to solve.

This article deals with all the information needed to understand the Derivative of the Logarithmic Function in plenty of detail including all the necessary formulas, and properties. We will also learn about the problem with their solutions as well as FAQs and practice problems on Differentiation of Log functions.

What are Logarithmic Functions?

The function that is the inverse of the exponential function is called the logarithmic function. It is represented as log_bx, where b is the base of the log. The value of x is the value which equals the base of the logarithm raised to a fixed number y, thus, the general form of a logarithmic function is:

y = log_bx

Where,

• y is the logarithm of x,
• b is the base of the logarithm, and 
• x is the input of the logarithm.

Note: As we know, logarithms and exponentials are related to each other such that If y = log_bx then, x = b^y.

Learn more about Logarithms.

Properties of Logarithmic Function

Some properties of the logarithmic function are listed below:

• log (XY) = log X + log Y
• log (X / Y) = log X - log Y
• log X^Y = Y log X
• log _YX = ln X / ln Y

Read more about Log Rules.

What is Derivative of Logarithmic Function?

The derivative of the logarithmic functions is solved using the properties of the logarithms and chain rule. It is used to solve complex functions which cannot be solved directly. By using the logarithmic functions, the complex functions can be simplified and can be evaluated easily.

Mostly, the exponential functions use the derivative of the logarithmic functions to get the solution of the complex functions. The functions of the form f(x)^g(x) can be easily evaluated using the derivative of the logarithms.

Derivative of Logarithmic Function Formula

There are three formulas for the derivatives of the logarithmic functions. The following are the formulas for the derivative of the logarithmic functions.

• Derivative of ln x
• Derivative of log_ax
• Derivative of ln f(x)

Let's discuss these formulas in detail.

Derivative of ln x

The derivative of ln x evaluates to the reciprocal of x. The formula for the derivative of ln x is given below:

(d / dx) ln x = 1 / x 

Where, x > 0

Derivative of log_ax

The derivative of log with base a and value x evaluates to the reciprocal of the product of x and ln a. The formula for the derivative of log_ax is given below:

(d/dx) log_ax = 1 / [x ln a] 

Where, a ≠ 1

Derivative of ln f(x)

The derivative of ln f(x) evaluates to the derivative of f(x) divided by f(x). The formula for the derivative of ln f(x) is given below:

(d/dx) ln f(x) = f'(x) / f(x)

Where,

• f(x) is any function of x, and
• f'(x) is derivative of function of x.

Proof of Derivative of Logarithmic Function Using First Principle

Using the first principle of derivative

(d / dx) f(x) = lim_h→0[{f(x +h) - f(x)} / {(x + h) - x}]

Here, f(x) = ln x

(d / dx) f(x) = lim_h→0[{ln(x +h) - ln(x)} / h]

⇒ (d / dx) f(x) = lim_h→0[{ln{(x +h)/(x)}} / h]

⇒ (d / dx) f(x) = lim_h→0[{ln(1 + (h / x))} / h]

Now, putting (h / x) = (1 /n) and as limit h→0 then, (1 / n) → ∞

(d / dx) f(x) = lim_n→∞ (n / x) [ln(1 + (1 / n))]

⇒ (d / dx) f(x) = lim_n→∞ (1 / x) ln(1 + (1 / n))ⁿ

The value of lim_n→∞ ln(1 + (1 / n))ⁿ = e

(d / dx) f(x) = (1 / x) ln e . . . (1)

⇒ (d / dx) ln x = 1 / x

For (d / dx) log_ax = 1 / (x ln a) putting in 1

(d / dx) log_ax = (1 / x) log_ae

⇒ (d / dx) log_ax = (1 / x) ln e /ln a [log_ae = ln e / ln a ]

⇒ (d / dx) log_ax = 1 / (x ln a)

Logarithmic Differentiation

Logarithmic Differentiation uses the chain rule of differentiation with the differentiation formula of the log, and it helps us differentiate complex functions with ease.

There are three forms of logarithmic differentiation i.e., differentiation of ln x, differentiation of log_ax and differentiation of ln f(x) whose differentiation formulas are mentioned above.

Let's consider an example for Logarithmic Differentiation.

Example: Find the derivative of log₃(x)

Solution:

Let f(x) = log₃(x)

⇒ f'(x) = (d /dx) [log₃x]

⇒ f'(x) = 1 / [xln 3]

Read more about, Logarithmic Differentiation.

Also,Check

• Inverse Trig Derivatives
• Differentiation and Integration Formula
• Derivative of Exponential Function

Solved Examples on Derivatives of Logarithmic Function

Example 1: Evaluate:

(i) Derivative of log 2

(ii) Derivative of log 3

(iii) Derivative of log 5

(iv) Derivative of log 10

Solution:

As Derivative of log 2, Derivative of log 3, Derivative of log 5, and Derivative of log 10 are all constant values, and derivative of any constant is 0.

Thus, Derivative of log 2, Derivative of log 3, Derivative of log 5, and Derivative of log 10 are all equals to 0.

Example 2: Find the following derivatives.

(i) Derivative of log 2x

(ii) Derivative of log₁₀x

(iii) Derivative of log y

Solution:

(i) Derivative of log2x

⇒ (d / dx) [log 2x] = (d / dx) [log 2x](d / dx) [2x]

⇒ (d / dx) [log 2x] = 2 / (2x)

⇒ (d / dx) [log 2x] = 1 / x

(ii) Derivative of log₁₀x

⇒ (d / dx) [log₁₀x] = 1 / [x ln 10]

(iii) Derivative of log y

⇒ (d / dx) [log y] = [1 / y](dy / dx)

Example 3: Find the derivative of ln (x² + 4)

Solution:

Let p(x) = ln(x² + 4)

⇒ p'(x) = (d / dx)[ln(x² + 4)]

By using formula

(d / dx) ln f(x) = f'(x) / f(x)

⇒ p'(x) = (2x) /(x² + 4)

Example 4: Evaluate: ln[(x²sinx) / (2x + 1)]

Solution:

Let f(x) = ln[(x²sinx) / (2x + 1)]

By using the properties of ln

f(x) = 2lnx + ln(sinx) - ln(2x +1)

Now, differentiating,

f'(x) = 2(d /dx)lnx + (d / dx)[ln(sinx)] - (d / dx)[ln(2x +1)]

⇒ f'(x) = (2 / x) + (cos x / sin x) - [2 / (2x +1)]

⇒ f'(x) = (2 / x) + cot x - [2 / (2x +1)]

Example 5: Find the slope of the line tangent of the graph of y = log₂(3x +1) at x = 1.

Solution:

The slope of line tangent of a graph is given by (dy / dx)

(dy / dx) = (d / dx) [log₂(3x +1)]

⇒ (dy / dx) = 3 / [(3x +1)ln2]

At x = 1

⇒ (dy / dx)_{at x =1} = 3 / [(3(1) +1)ln2]

⇒ (dy / dx)_{at x =1} = 3 / [4 ln2]

⇒ (dy / dx)_{at x =1} = 3 / [ln2⁴]

⇒ (dy / dx)_{at x =1} = 3 / [ln16]

Example 6: Evaluate the derivative: y = (2x⁴ + 1)^tanx.

Solution:

Taking logarithm both sides

ln y = ln (2x⁴ + 1)^tanx

⇒ ln y = tanx [ln (2x⁴ + 1)]

Differentiating

(d / dx)ln y = (d / dx)[tanx [ln (2x⁴ + 1)]]

Applying formula and product rule

(1 / y) (dy / dx) = [(d / dx)tanx {ln (2x⁴ + 1)}] + [tan x (d / dx){ln (2x⁴ + 1)}]

⇒ (1 / y) (dy / dx) = [sec²x ln (2x⁴ + 1)] + [tan x {8x³ / (2x⁴ + 1)}]

⇒ (dy / dx) = (2x⁴ + 1)^tanx[sec²x ln (2x⁴ + 1) + tan x {8x³ / (2x⁴ + 1)}]

Example 7: Find the derivative y = x^x.

Solution:

y = x^x

Taking logarithm both sides

ln y = ln x^x

By logarithm properties

ln y = x lnx

Differentiating

(d /dx) ln y = (d /dx) [x ln x]

Applying product rule

(1 /y) (dy /dx) = (d /dx) (x) (ln x) + x(d /dx)( ln x)

⇒ (1 /y) (dy /dx) = ln x + x( 1/x)

⇒ (dy /dx) = y(ln x + 1)

⇒ (dy /dx) = x^x(ln x + 1)

Practice Problems on Differentiation of Logarithmic Function

Problem 1: Calculate the derivative of g(x) = 3ln(2x).

Problem 2: Determine the derivative of h(x) = ln(5x²).

Problem 3: Find the derivative of p(x) = ln(4x³ + 2x).

Problem 4: Calculate the derivative of q(x) = ln(1/x).

Problem 5: Determine the derivative of r(x) = ln(3x² - 7x + 1).

Problem 6: Find the derivative of s(x) = 2ln(sqrtx).

Problem 7: Calculate the derivative of t(x) = ln(x³ + 4x² - 2x + 1).

Problem 8: Determine the derivative of u(x) = ln(e^x + 1).

Problem 9: Find the derivative of v(x) = ln(2x³ - 3x² + 5x - 7).