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⁄dₓ, 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 lneu, write 1 in numerator and take the argument u in denominator. By chain rule, further derivative of u with respect to x is ᵈᵘ⁄dₓ.
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(sin√x)
Solution :
y = ln(sin√x)
Kindly mail your feedback to v4formath@gmail.com
We always appreciate your feedback.
©All rights reserved. onlinemath4all.com
May 26, 23 12:27 PM
May 21, 23 07:40 PM
May 20, 23 10:53 PM