Question

How do you differentiate $4xy-3x-11=0$?

Answer 1

To solve the given question we will use the concept of implicit differentiation. We will differentiate the given equation with respect to x. In order to differentiate the equation we will use the product rule and power rule of the differentiation.

Complete step by step solution:
We have been given an equation $4xy-3x-11=0$.
We have to differentiate the given equation.
Now, differentiating the given equation with respect to x we will get
$\Rightarrow \dfrac{d}{dx}\left( 4xy \right)-\dfrac{d}{dx}3x-\dfrac{d}{dx}11=\dfrac{d}{dx}0$
Now, we know that $\dfrac{d}{dx}\left( uv \right)=u\dfrac{d}{dx}v+v\dfrac{d}{dx}u$ and $\dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}$
Now, applying the differentiation rules to the above obtained equation we will get
$\Rightarrow 4\left( x\dfrac{d}{dx}y+y\dfrac{d}{dx}x \right)-3-0=0$
Now, simplifying the above obtained equation we will get
$\begin{aligned} & \Rightarrow 4\left( x\dfrac{dy}{dx}+y \right)-3=0 \\\ & \Rightarrow 4x\dfrac{dy}{dx}+4y=3 \\\ & \Rightarrow 4x\dfrac{dy}{dx}=3-4y \\\ \end{aligned}$
Now, to find the value of $\dfrac{dy}{dx}$ we need to rearrange the terms in the above obtained equation. Then we will get
$\Rightarrow \dfrac{dy}{dx}=\dfrac{3-4y}{4x}$
Hence above is the required solution of the given equation.

Note: In this particular question we are assuming that the y is the function of x so we use the concept of implicit differentiation to solve further. If students consider the y as a constant and solve accordingly they will get the incorrect solution. Alternatively we can simplify the equation and convert it into the function of x alone and then differentiate the equation.
We can rewrite the given equation as
$\begin{aligned} & \Rightarrow 4xy-3x-11=0 \\\ & \Rightarrow 4xy=11+3x \\\ & \Rightarrow y=\dfrac{3x+11}{4x} \\\ \end{aligned}$
Now, using the quotient rule of the differentiation we can solve the equation and get the desired answer.