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

1 Answer
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(x,y) = 0

=> 7xy−3lny-42=0

Differentiation w.r.t. x on both sides, we get:
d/dx(7xy)-3d/dx(lny) = 0
=>7x . dy/dx + 7y -3*dy/dx*d/dy(lny) = 0
=> 7x . dy/dx + 7y -3*dy/dx*1/y = 0
=> dy/dx*(7x-3/y)+7y =0
=>dy/dx=-(7*y)/(7x-3/y)
=>(dy/dx)_(x,y)=(7*y)/(3/y-7x)

Now, we have determined the general equation for the 1st derivative for the given function. We just need to put the values of (x,y) = (6,1) in the given equation and we are all set!
Substituting, (dy/dx)_(6,1) = 7*1/(3/1-6*7)
=>dy/dx = -7/39