Answer

Verified

436.8k+ views

**Hint:**The given function has two independent functions of x in product form. Since both are x dependent thus they can’t be directly differentiated with respect to x. These type of problems are solved by a particular method using product rule, given by \[\dfrac{\text{dy}}{\text{dx}}=\text{u}\times \dfrac{\text{d}}{\text{dx}}(v)+\text{v}\times \dfrac{d}{\text{dx}}\left( u \right)\], where function is \[\text{y=u}\times \text{v}\]. We will also be using standard derivatives like \[\dfrac{\text{d}}{\text{dx}}\left( \text{sinx} \right)=\text{cosx},\text{ }\dfrac{\text{d}}{\text{dx}}\left( \text{cosx} \right)=\text{-sinx}\], \[\dfrac{\text{d}}{\text{dx}}(\text{lnx)=}\dfrac{1}{\text{x}}\] to solve the question.

**Complete step-by-step solution:**Let us learn about the product rule first. In product rule, we have two functions of x, assigning them as first and second functions respectively. We keep the first function constant and differentiate the second function with respect to x then, keeping the second function constant and differentiate the first function with respect to x.

Suppose, \[\text{y=u}\times \text{v}\]; u = first function of x and v = second function of x.

Then, product rule is given by \[\dfrac{\text{dy}}{\text{dx}}=\text{u}\times \dfrac{\text{d}}{\text{dx}}(v)+\text{v}\times \dfrac{d}{\text{dx}}\left( u \right)\].

Here we have the function $y=\text{(sinx)}\text{.lnx}$.

The given function has two independent functions $\sin x$ and $\ln x$. So, y cannot be differentiated directly.

As we have already discussed we use chain rule for solving these types of particular questions. Here, we have the first function as $\sin x$ and the second function as $\ln x$.

Therefore, we can write it as \[\dfrac{\text{dy}}{\text{dx}}=\dfrac{\text{d}}{\text{dx}}\left( \left( \text{sinx} \right).\text{lnx} \right)\]

Now, applying the product rule, we get

\[\Rightarrow \dfrac{\text{dy}}{\text{dx}}=\text{lnx}\text{.}\dfrac{\text{d}}{\text{dx}}\left( \text{sinx} \right)+\text{sinx}\text{.}\dfrac{\text{d}}{\text{dx}}\left( \ln x \right)\]

Since, we know that \[\dfrac{\text{d}}{\text{dx}}\left( \text{sinx} \right)=\text{cosx and }\dfrac{\text{d}}{\text{dx}}\left( \text{lnx} \right)=\dfrac{1}{\text{x}}\]

On putting these derivatives, we get:

\[\begin{align}

& \Rightarrow \dfrac{\text{dy}}{\text{dx}}=\text{lnx}\text{.cosx+sinx}\text{.}\dfrac{1}{\text{x}} \\

& \Rightarrow \dfrac{\text{dy}}{\text{dx}}=\dfrac{1}{\text{x}}\left( \text{x}\text{.lnc}\text{.cosx+sinx} \right) \\

\end{align}\]

**Therefore, option D is correct.**

**Note:**Students may directly differentiate the y and give the answer as \[\dfrac{\text{d}}{\text{dx}}\left( \left( \text{sinx} \right)\text{lnx} \right)=\dfrac{\text{cosx}}{\text{x}}\] which is totally wrong. So this kind of silly mistake must be avoided. Also take the derivative of functions sinx and cosx properly, without making sign changes.

Recently Updated Pages

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Advantages and disadvantages of science

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Difference Between Plant Cell and Animal Cell

10 examples of evaporation in daily life with explanations

Give 10 examples for herbs , shrubs , climbers , creepers

Write a letter to the principal requesting him to grant class 10 english CBSE

How do you graph the function fx 4x class 9 maths CBSE