# DERIVATIVE OF LOGARITHMIC FUNCTIONS

Consider the following logarithmic function.

y = ln[f(x)]

In a logarithm, if we have the spelling "ln", its a natural logarithm and its base is 'e'.

So, ln[f(x)] is a natural logarithm and its base is e.

Then, we have

y = lne[f(x)]

In lne[f(x)],

base = e

argument = f(x)

## Derivative of a Natural Logarithmic Function

Wroking rule to find the derivative of a function which contains natural logarithm in the form y = ln[f(x)].

In y = ln[f(x)], since y is a function of x, we can find the derivative of y with respect to x, that is ᵈʸ⁄d.

Step 1 :

y = ln[f(x)]

Let t = f(x).

Then, we have

y = ln(t)

Step 2 :

Now, y = ln(t) and t = f(x).

y = ln(t) ----> y is a function of t

t = f(x) ----> t is a function of x

Step 3 :

In step 2,

(i) since y is a function of t, we can find the derivative of y with respect to t.

(i) since t is a function of x, we can find the derivative of t with respect to x.

Chain rule to find derivative of y with respect to x.

Step 4 :

Substitute y = ln(t) and t = f(x).

To get the derivative of ln(t) with respect to t, write 1 in numerator and take the argument u in denominator. And the derivative f(x) with respect to x is f'(x).

Substitute t = f(x).

## Derivative of a General Logarithmic Function

Working rule to find the derivative of a logarithmic function where the logarithm is not a natural logarithm :

Consider the following equation.

y = loga[f(x)]

Step 1 :

Convert the above equation to exponential form.

ay = f(x)

Step 2 :

Take natural logarithm on both sides.

ln(ay) = ln[f(x)]

Step 3 :

Use the power rule of logarithm.

yln(a) = ln[f(x)]

Step 4 :

Find the derivative on both sides.

## Solved Problems

Find ᵈʸ⁄dₓ in each of the following.

Problem 1 :

y = ln(x)

Solution :

y = ln(x)

Let t = x.

y = ln(t)

y = ln(t) ----> y is a function of t

t = x ----> t is a function of x

Chain Rule :

Substitute y = ln(t) and t = x.

Substitute t = x.

Problem 2 :

y = ln(2x)

Solution :

y = ln(2x)

Let t = 2x.

y = ln(t)

y = ln(t) ----> y is a function of t

t = 2x ----> t is a function of x

Chain Rule :

Substitute y = ln(t) and t = 2x.

Substitute t = 2x.

Problem 3 :

y = ln(x2 + 5x - 6)

Solution :

y = ln(x2 + 5x - 6)

Let t = x2 + 5x - 6.

y = ln(t)

y = ln(t) ----> y is a function of t

t = x2 + 5x - 6 ----> t is a function of x

Chain Rule :

Substitute y = ln(t) and t = x2 + 5x - 6.

Substitute t = x2 + 5x - 6.

Problem 4 :

y = ln(ex)

Solution :

y = ln(ex)

Let t = ex.

y = ln(t)

y = ln(t) ----> y is a function of t

t = ex ----> t is a function of x

Chain Rule :

Substitute y = ln(t) and t = ex.

Substitute t = ex.

Problem 5 :

y = ln(√x)

Solution :

y = ln(√x)

Let t = √x.

y = ln(t)

y = ln(t) ----> y is a function of t

t = √x ----> t is a function of x

Chain Rule :

Substitute y = ln(t) and t = √x.

Substitute t = √x.

Problem 6 :

y = ln[sin(x)]

Solution :

y = ln[sin(x)]

Let t = sin(x).

y = ln(t)

y = sin(x) ----> y is a function of t

t = √x ----> t is a function of x

Chain Rule :

Substitute y = ln(t) and t = sin(x).

Substitute t = sin(x).

Problem 7 :

y = ln[cos(√x)]

Solution :

y = ln[cos(√x)]

Let u = √x.

y = ln[cos(u)]

Let v = cos(u).

y = ln(v)

y = ln(v) ----> y is a function of v

v = cos(u) ----> v is a function of u

u = √x ----> u is a function of x

Chain Rule :

Substitute y = ln(v), v = cos(u) and u = √x.

Substitute v = cos(u).

Substitute u = √x.

Problem 8 :

Solution :

Let u = tan(x).

y = ln(u)

Let v = u.

y = ln(v)

y = ln(v) ----> y is a function of v

v = √u ----> v is a function of u

u = tan(x) ----> u is a function of x

Chain Rule :

Substitute y = ln(v), v = u and u = tan(x).

Substitute v = u.

Substitute u = tan(x).

Problem 9 :

y = log10(x)

Solution :

y = log10(x)

Convert the above equation to exponential form.

10y = x

Take natural logarithm on both sides.

ln(10y) = ln(x)

Use the power ruole of logarithm.

yln(10) = ln(x)

Find the derivative on both sides with respect to x.

Problem 10 :

y = log6(x2 + 3x + 5)

Solution :

y = log10(x2 + 3x + 5)

Convert the above equation to exponential form.

6yx2 + 3x + 5

Take natural logarithm on both sides.

ln(6y) = ln(x2 + 3x + 5)

Use the power ruole of logarithm.

yln(6) = ln(x2 + 3x + 5)

Find the derivative on both sides with respect to x.

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