Determine the derivative

Question

What is the derivative of this and how do I find it?

Answer 1

It depends.

The derivative with respect to what variable?

It's usually with respect to x

Then the derivative is found by implicit differentiation with the chain rule

Left side's derivative would be 4 -12y^2(dy/dx) as y is treated as a function of x, y=f(x)

Right side's derivative would be 2xy(dy/dx) + y^2

It's the derivative of a product, which is the derivative of the first factor times the 2nd factor plus

the derivative of the 2nd factor times the 1st factor

That leads to

4-12y^2(dy/dx) = 2xy(dy/dx) + y^2

That may be all you want, or you can go on to solve for dy/dx

-12y^2(dy/dx) - 2xy(dy/dx) = y^2-4

(dy/dx)(-12y^2-2xy) = y^2-4

dy/dx = (y^2-4)/(-12y^2-2xy)

=(y^2-4)/2y(-6y-x)

or eliminate that unnecessary negative signs in the denominator

=(4-y^2)/2y(6y+x)

or you could factor the numerator

=(4-y)(4+y)/2y(6y+x)

the derivative is the slope of the graph of the equation

by finding values of x or y that make either the numerator or denominator zero

you can find where the graph is horizontal or approaches an infinite slope

but possibly you want the derivative with respect to time or t or some third variable

then the derivative

is 4(dx/dt) - 12y(dy/dt) = x(2y)(dy/dt) + y^2(dx/dt)

or 4x' -12yy' = 2xyy' +y^2y'

where x'=dx/dt and y'=dy/dt

It's just 2 different notations for the derivative , the dy/dt notation makes it clear what variable you're taking the derivative with respect to.

Answer 2

4x - 6y² = xy²

4 - 12y dy/dx = y² +x 2y dy/dx

dy/dx (2xy+12y) = y²- 4

dy/dx = (4 - y² ) / 2y( x +6)

replace y = √4x/(x+6)