Question

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

Answer 1

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`