DERIVATIVE OF A NATURAL LOGARITHMIC FUNCTION

Consider the following logarithmic function.

y = lnu

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

So, lnu is a natural logarithm and its base is e.

Then, we have

y = lneu

In lneu,

base = e

argument = u

Wroking rule to find the derivative of a natural logarithmic function :

Find ᵈʸ⁄d, if y = lneu, where u is a function of x.

Step 1 :

y = lneu

Convert the above equation to exponential form.

ey = u

Step 2 :

Take natural logarithm on both sides.

lneey = lneu

Step 3 :

Use the power rule of logarithm.

ylnee = lneu

y(1) = lneu

y = lneu

Step 4 :

Find the derivative on both sides.

To get the derivative of lneuwrite 1 in numerator and take the argument u in denominator. By chain rule, further derivative of u with respect to x is ᵈᵘ⁄d.

Solved Problems

Find ᵈʸ⁄dₓ in each of the following.

Problem 1 :

y = ln(x)

Solution :

y = ln(x)

Problem 2 :

y = ln(2x)

Solution :

Method 1 :

y = ln(2x)

Method 2 :

y = ln(2x)

Use the product rule of logarithm.

y = ln(2) + ln(x)

Problem 3 :

y = ln(5x3)

Solution :

Method 1 :

y = ln(5x3)

Method 2 :

y = ln(5x3)

Use the product rule of logarithm.

y = ln(5) + ln(x3)

Use the power rule of logarithm.

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

Problem 4 :

y = ln(x2 - 3x + 6)

Solution :

y = ln(x2 - 3x + 6)

Problem 5 :

y = ln(x)

Solution :

Method 1 :

y = ln(x)

Method 2 :

y = ln(x)

y = ln(x½)

Use the power rule of logarithm.

Problem 6 :

y = ln(ex)

Solution :

Method 1 :

y = ln(ex)

Method 2 :

y = ln(ex)

Use the power rule of logarithm.

y = xln(e)

y = xlnee

y = x(1)

y = x

Problem 7 :

y = ln(sinx)

Solution :

y = ln(sinx)

Problem 8 :

y = ln(cosx)

Solution :

y = ln(cosx)

Problem 9 :

y = ln(tanx)

Solution :

y = ln(tanx)

Problem 10 :

y = ln(sinx)

Solution :

y = ln(sinx)

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