Help with function

SmokyMo

The equation x^3y^2 + lny^4 = 8 determines an implicit function : x - y(x).
Find the derivative, y'(x), of this function at the point (2; 1).

First i should find a derivative which is (3x^2)(2y)+4y^3=0(is it right?)
then what should i do?

Michael Collins

You have to differentiate it implicitly. For the first term you'll have to do something like:

Let u = x^f(y),
ln u = ln(x^f(y)) = f(y)*ln(x)

differentiate both sides:

1/u du/dx = f(y).1/x + ln(x).f'(y) ...just implicit differentiation on the LHS and the product rule on the RHS

du/dx = u*(f(y).1/x + ln(x).f'(y))

so apply that to the first term, then the second terms becomes

d/dx [ln(y^4)] = 1/(y^4) . 4*y^3 . dy/dx

Now just sub in the values for x and y i.e. 2 and 1 respectively. Then solve for dy/dx.

SmokyMo

x^3y^2 + lny^4 = 8 - the equation

differentiated

3x^2y^2 + 2x^3y dy/dx + 4/y dy/dx = 0
3x^2y^2 +(2x^3y + 4/y) dy/dx = 0

dy/dx=(3x^2y^2)/(2x^3y + 4/y)
plug in the values (2,1)
and I get -3/5

Is it right?