Question

How do you use the logarithmic differentiation to find the derivative of a function.

Answer 1

Hint:
We will first define what is logarithmic differentiation and why do we use it and then we will try to understand how logarithmic differentiation simplify the differentiation of a function with the help of some example.

Complete step by step answer:
We will first understand what is logarithmic differentiation. Logarithmic differentiation is basically taking the logarithmic of the function given and then differentiating the function obtained after taking logarithmic so that it simplifies the differentiation of the original function.
We generally use logarithmic differentiation to simplify the differentiation of the complicated function.
Now, we will under the method of logarithmic differentiation with the help of examples.
Example 1: We can find the differentiation of $ y=\dfrac{{{\left( 3x+5 \right)}^{5}}}{{{\left( 4x-7 \right)}^{7}}} $ by using the quotient rule, the chain rule and the power rule and then simplify it algebraically but that procedure is too lengthy. So, we will try to apply logarithmic differentiation here.
We will first find the logarithm of the given function by taking log on both the side of the equation:
 $ \Rightarrow ~\log y=\log \left( \dfrac{{{\left( 3x+5 \right)}^{5}}}{{{\left( 4x-7 \right)}^{7}}} \right) $
 $ \Rightarrow ~\log y=\log \left( {{\left( 3x+5 \right)}^{5}} \right)-\log \left( {{\left( 4x-7 \right)}^{7}} \right) $ , because we know that $ \log \left( \dfrac{a}{b} \right)=\log a-\log b $
 $ \Rightarrow \log y=5\log \left( 3x+5 \right)-7\log \left( 4x-7 \right) $ , because we know that $ \log {{a}^{b}}=b\log a $
Now, upon differentiating both the sides with respect to x we will get:
 $ \Rightarrow \dfrac{1}{y}\dfrac{dy}{dx}=\dfrac{5\times 3}{\left( 3x+5 \right)}-\dfrac{7\times 4}{\left( 4x-7 \right)} $ , since we know that $ \dfrac{d\left( \log x \right)}{dx}=\dfrac{1}{x} $ and we have also applied chain rule here.
 $ \Rightarrow \dfrac{dy}{dx}=y\left( \dfrac{5\times 3}{\left( 3x+5 \right)}-\dfrac{7\times 4}{\left( 4x-7 \right)} \right) $
 $ \Rightarrow \dfrac{dy}{dx}=y\left( \dfrac{15}{\left( 3x+5 \right)}-\dfrac{28}{\left( 4x-7 \right)} \right) $
Since, we know that $ y=\dfrac{{{\left( 3x+5 \right)}^{5}}}{{{\left( 4x-7 \right)}^{7}}} $ , so we will put it in above equation: $ \Rightarrow \dfrac{dy}{dx}=\dfrac{{{\left( 3x+5 \right)}^{5}}}{{{\left( 4x-7 \right)}^{7}}}\left( \dfrac{15}{\left( 3x+5 \right)}-\dfrac{28}{\left( 4x-7 \right)} \right) $
This is our required answer of the above equation.

Note:
Students are required to note that when we have to find the differentiation of the function which has higher power or the function in which we have a variable in the power, then generally logarithmic differentiation simplifies our work and we will be able to find the derivative more easily of such complicated function.