# How do you differentiate 7xy- 3 lny= 42 at the point (6,1)?

Jun 5, 2015

We have to solve the given equation by implicit differentiation.

The given equation is 7xy−3lny=42.
Let us express it in the form of $F \left(x , y\right) = 0$

=> 7xy−3lny-42=0

Differentiation w.r.t. x on both sides, we get:
$\frac{d}{\mathrm{dx}} \left(7 x y\right) - 3 \frac{d}{\mathrm{dx}} \left(\ln y\right) = 0$
$\implies 7 x . \frac{\mathrm{dy}}{\mathrm{dx}} + 7 y - 3 \cdot \frac{\mathrm{dy}}{\mathrm{dx}} \cdot \frac{d}{\mathrm{dy}} \left(\ln y\right) = 0$
$\implies 7 x . \frac{\mathrm{dy}}{\mathrm{dx}} + 7 y - 3 \cdot \frac{\mathrm{dy}}{\mathrm{dx}} \cdot \frac{1}{y} = 0$
$\implies \frac{\mathrm{dy}}{\mathrm{dx}} \cdot \left(7 x - \frac{3}{y}\right) + 7 y = 0$
$\implies \frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{7 \cdot y}{7 x - \frac{3}{y}}$
$\implies {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}_{x , y} = \frac{7 \cdot y}{\frac{3}{y} - 7 x}$

Now, we have determined the general equation for the 1st derivative for the given function. We just need to put the values of $\left(x , y\right) = \left(6 , 1\right)$ in the given equation and we are all set!
Substituting, ${\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}_{6 , 1} = 7 \cdot \frac{1}{\frac{3}{1} - 6 \cdot 7}$
$\implies \frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{7}{39}$