Steps involved in logarithmic differentiation :
Step 1 :
Take natural logarithm on both sides.
Step 2 :
Use the properties of logarithms.
(i) The Product Rule
(ii) The Quotient Rule
(iii) The Power Rule
Step 3 :
Find derivative.
In calculus, when we find derivative of a function, we will be using logarithmic differentiation in the following two cases.
Case (1) :
y = 3^{x}
In the function above, we have variable x at the exponent. To find derivative of this kind of function (a function containing variable at the exponent), we must use logarithmic differentiation.
Case (2) :
To find the derivative of the function given above, first, we have to write the square root as exponent. Then, the exponent will be ½. Now, we have to find the derivative for the exponent and find derivative for the stuff inside the square root using chain rule. When we find the derivative of the stuff inside the square root, we have to use the quotient rule. Because, both numerator and denominator contain variable.
If we find the derivative of the above function as it is, the process of derivative will be litter longer.
Whenever we want to find derivative of this kind of complicated function, we can use logarithmic differentiation. When we use logarithmic differentiation, we can split up the function into parts using the properties of logarithm and make the derivative process easier.
Problem 1 :
y = 7^{x}
Solution :
y = 7^{x}
Take natural logarithm on both sides.
ln y = ln (7^{x})
Use the power rule of logarithm on the right side.
ln y = xln 7
Find the derivative on both sides with respect to x (use the chaion rule of derivative on the left side).
Problem 2 :
Solution :
Take natural logarithm on both sides.
Use the properties of logarithms.
Find the derivative on both sides with respect to x (use the chaion rule of derivative on the left side).
Problem 3 :
Solution :
Take natural logarithm on both sides.
Use the power rule of logarithm on the right side.
Find the derivative on both sides with respect to x (use the chain rule of derivative on the left side and the product rule of derivative on the right side).
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