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 = ln_eu

In ln_eu,

base = e

argument = u

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(x^½)

Use the power rule of logarithm.

Problem 6 :

y = ln(e^x)

Solution :

Method 1 :

y = ln(e^x)

Method 2 :

y = ln(e^x)

Use the power rule of logarithm.

y = xln(e)

y = xln_ee

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(sin√x)

Solution :

y = ln(sin√x)

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