Question

How do you use the product rule to find the derivative of \[y = x \cdot \ln \left( x \right)\] ?

Answer 1

Hint: In the given question we have to find the derivative of \[y = x \cdot \ln \left( x \right)\] using the product rule. Now for this you should know what the product rule is. So the product rule is used when you have to find the derivative of a product of two or more functions. Here we have mention the product rule formula for two functions which you have to use to find the solution of the given question i.e.
\[\dfrac{d}{{dx}}\left( {u \cdot v} \right) = u\dfrac{d}{{dx}}\left( v \right) + v\dfrac{d}{{dx}}\left( u \right)\]
Here two functions were \[u(x)\] and \[v(x)\]
Now first find what are the two different functions in the given term and then apply the above given rule and find the derivative. And be careful while differentiating different functions.

Complete step by step solution:
In the given question we have to use the product rule to find the derivative of \[y = x \cdot \ln \left( x \right)\] so here we will first find what the two different functions are in the given term and then apply the product rule.
So the two function are \[x\] and \[\ln \left( x \right)\]
Now let use the product formula on these functions i.e.
 \[
  \dfrac{d}{{dx}}\left( {u \cdot v} \right) = u\dfrac{d}{{dx}}\left( v \right) + v\dfrac{d}{{dx}}\left( u \right) \\
  \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {x \cdot \ln x} \right) \;
 \]
Now differentiate the following term we with respect to \[x\] we get,
 \[
  \dfrac{{dy}}{{dx}} = x\left( {\dfrac{1}{x}} \right) + \ln x\left( 1 \right) \\
  \dfrac{{dy}}{{dx}} = 1 + \ln x \;
 \]
Hence, \[(1 + \ln x)\] this is the required answer of the given question.
So, the correct answer is “ \[(1 + \ln x)\] ”.

Note: Here first of all you should know the differentiations of simple functions learn all the derivatives of basic functions. Now you have to be careful while doing the differentiation because most of the students make mistakes while differentiating the terms. Also be careful while separating different functions available in the given term. You should also search the proof of the product rule.