Question

How do you differentiate \[y = \log {x^2}\] ?

Answer 1

Hint: To solve the question given above, you need to know about logarithmic derivation. The logarithmic derivative of the initial function \[y = f\left( x \right)\] is the derivative of the logarithmic function. The derivatives of power-exponential functions, or functions of the form, can be computed effectively using this differentiation method. Where \[u\left( x \right)\] and \[v\left( x \right)\] are differentiable functions of \[x\], \[y = u\left( x \right)v\left( x \right)\].

Formula used:
We will be using two approaches to solve the question mentioned above. The formulas required to solve the question using two different approaches are:
We can solve this question using the properties of logarithm. We know that \[\log {x^n} = n\log x\].
In the second approach we will use the chain rule where: \[\dfrac{d}{{dx}}\left( {\log \left( {f\left( x \right)} \right)} \right) = \dfrac{1}{{f\left( x \right)}} \times f'\left( x \right)\]

Complete step-by-step answer:
We are given: \[y = \log {x^2}\]
We will differentiate this function by using the laws of logarithm.
Now, using the first formula: \[\log {x^n} = n\log x\]
We get: \[y = 2\log x\]
On differentiating both sides with respect to x, we get:
\[\dfrac{{dy}}{{dx}} = 2 \times \dfrac{1}{x}\]
       \[ = \dfrac{2}{x}\].
So, our final answer is \[\dfrac{2}{x}\]
Additional information:
This question can also be solved using the chain rule:
We know that: \[\dfrac{d}{{dx}}\left( {\log \left( {f\left( x \right)} \right)} \right) = \dfrac{1}{{f\left( x \right)}} \times f'\left( x \right)\]
On differentiating \[y = \log {x^2}\] using the chain rule, we get:
\[\dfrac{{dy}}{{dx}} = \dfrac{1}{{{x^2}}} \times \dfrac{d}{{dx}}\left( {{x^2}} \right)\]
       \[
   = \dfrac{1}{{{x^2}}} \times 2x \\
   = \dfrac{2}{x} \\
 \]
The final answer is \[\dfrac{2}{x}\].

Note: While solving questions similar to the one given above, remember the chain rule is a method for determining the derivative of composite functions, with the number of functions in the composition determining the number of differentiation steps required. Consider an example: \[f\left( x \right) = \left( {goh} \right)\left( x \right) = g\left( {h\left( x \right)} \right)\]then this becomes \[f'\left( x \right) = g'\left( {h\left( x \right)} \right) \times h'\left( x \right)\]. Since the composite function f is made up of two functions, g and h, you must differentiate \[f\left( x \right)\] using the derivatives \[g'\] and \[h'\].