Question

What is the differentiation of \[uv\]?

Answer 1

Hint: Differentiation of \[uv\] can be obtained by the product rule. Product rule is the formula used to find the derivatives of a product of two or more functions. if \[u,v\] are two functions of \[x\], then the derivative of product of \[uv\] is given by \[\dfrac{d}{dx}\left( uv \right)=u\dfrac{d}{dx}\left( v \right)+v\dfrac{d}{dx}\left( u \right)\].

Complete step-by-step solution:
From the question it is clear that we have to write the differentiation of \[uv\]. Which we can simply write from the product rule.
Let us assume \[u,v\] are two functions of \[x\].
Product rule is the formula used to find the derivatives of a product of two or more functions.
When the first function is multiplied by the derivative of the second function plus the second function multiplied by the derivative of the first function is known as product rule. This is an easy way of remembering and easy writing.
Let us assume \[uv\] as \[y\]
\[\Rightarrow y=uv\]……………(1)
From the question we were asked to find the derivative of \[uv\]. So, finding \[\dfrac{dy}{dx}\] is equal to finding \[\dfrac{d}{dx}\left( uv \right)\].
So,
\[\Rightarrow \dfrac{dy}{dx}=\dfrac{d}{dx}\left( uv \right)\]……………….(2)
Using product rule, we can write the equation (2) as
\[\Rightarrow \dfrac{d}{dx}\left( uv \right)=u\dfrac{d}{dx}\left( v \right)+v\dfrac{d}{dx}\left( u \right)\]
So finally, we can conclude that derivative of \[uv\]is \[u\dfrac{d}{dx}\left( v \right)+v\dfrac{d}{dx}\left( u \right)\].

Note: Students should be careful while applying product rules. Small errors in the application of product rule may lead to doing this question wrong and may get the wrong answer. Many students may have misconception that \[\dfrac{d}{dx}\left( uv \right)=\dfrac{d}{dx}\left( u \right)\times \dfrac{d}{dx}\left( v \right)\], which is completely wrong formula. Using this wrong formula may lead to this question wrong.