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

Jul 16, 2015

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 - 4 y}{4 x}$

#### Explanation:

Assume the equation $4 x y - 3 x - 11 = 0$ implicitly defines $y$ as a function of $x$ and then differentiate with respect to $x$, using the Product Rule:

$4 y + 4 x \frac{\mathrm{dy}}{\mathrm{dx}} - 3 = 0$

Now solve for $\frac{\mathrm{dy}}{\mathrm{dx}}$ to get $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 - 4 y}{4 x}$.

If you happen to know a specific point $\left(x , y\right)$ on the curve defined by the original equation $4 x y - 3 x - 11 = 0$, you can plug the coordinates of that point into $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 - 4 y}{4 x}$ to find the slope of the curve at that point.

The particular equation $4 x y - 3 x - 11 = 0$ is simple enough that we can check our work another way; we can solve for $y$ explictly as a function of $x$:

$4 x y - 3 x - 11 = 0 \setminus R i g h t a r r o w y = \frac{3 x + 11}{4 x} = \frac{3}{4} + \frac{11}{4} {x}^{- 1}$

Now we can differentiate this in the ordinary way to get $\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{11}{4} {x}^{- 2} = - \frac{11}{4 {x}^{2}}$. Is this answer the same as the original? It sure looks different. It can be seen to be the same answer by substituting $y = \frac{3 x + 11}{4 x}$ into $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 - 4 y}{4 x}$ in place of $y$ and simplifying:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 - 4 \left(\frac{3 x + 11}{4 x}\right)}{4 x} = \frac{3 x - \left(3 x + 11\right)}{4 {x}^{2}} = - \frac{11}{4 {x}^{2}}$

It is the same!

As an example of a point on this curve, we can use the equation $y = \frac{3 x + 11}{4 x}$ to find $y$ when $x = 1$ to be $y = \frac{14}{4} = \frac{7}{2}$, meaning that the point $\left(x , y\right) = \left(1 , \frac{7}{2}\right)$ is on the graph of $4 x y - 3 x - 11 = 0$. The slope of the curve at that point is $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 - 14}{4} = - \frac{11}{4} = - 2.75$.

The equation of the tangent line to the curve at that point is therefore $y = - \frac{11}{4} \left(x - 1\right) + \frac{7}{2} = - \frac{11}{4} x + \frac{25}{4}$.

Here's a graph of the situation just described:

Jul 16, 2015

Assuming that we want to find $\frac{\mathrm{dy}}{\mathrm{dx}}$:

Using implicit differentiation, we get: $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 - 4 y}{4 x}$.

Solving for $y$ first, we get: $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{11}{4 {x}^{2}}$.

#### Explanation:

The question is posted under "Implicit Differentiation", so let's do it that way first:

$4 x y - 3 x - 11 = 0$

Leaving the function Implicit
In order to differentiate $4 x y$, we will need the product rule.

Remember that we are assuming that $y$ is some function of $x$, so we have $4 x y = 4 x f \left(x\right)$ and we use the product rule to get:
the derivative is: $4 f \left(x\right) + 4 x f ' \left(x\right)$

Back to this problem:

$\frac{d}{\mathrm{dx}} \left(4 x y\right) - \frac{d}{\mathrm{dx}} \left(3 x\right) - \frac{d}{\mathrm{dx}} \left(11\right) = \frac{d}{\mathrm{dx}} \left(0\right)$

$4 y + 4 x \frac{\mathrm{dy}}{\mathrm{dx}} - 3 = 0$

$4 x \frac{\mathrm{dy}}{\mathrm{dx}} = 3 - 4 y$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 - 4 y}{4 x}$

Making the function explicit

Solve $4 x y - 3 x - 11 = 0$ for $y$

$y = \frac{3 x + 11}{4 x}$

We could differentiate using the quotient rule, but it is perhaps simpler to rewrite again:

$y = \frac{3 x}{4 x} + \frac{11}{4 x}$

$= \frac{3}{4} + \frac{11}{4} {x}^{-} 1$

So
$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{11}{4} {x}^{-} 2$

$= - \frac{11}{4 {x}^{2}}$

To see that the answer are equivalent compare:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 - 4 y}{4 x}$

with

$y = \frac{3 x + 11}{4 x}$ and $\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{11}{4 {x}^{2}}$

Using Implicit differentiation, there is still a $y$ in the derivative. That is the price we pay for not making the function explicit before differentiating. If we substitute the solution for $y$, we get:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 - 4 \left(\frac{3 x + 11}{4 x}\right)}{4 x}$

$= \frac{3 - \frac{3 x + 11}{x}}{4 x}$

$= \frac{3 x - 3 x - 11}{4 {x}^{2}}$

$= - \frac{11}{4 {x}^{2}}$