CALCULUS Differentiation?

Question

possibly 2 tough derivative questions:
Determine the first derivative.

y(sqaureroot of x) - x(squareroot of y) = 16

and

2/3sin^3/2x - 2/7sin^7/2x
these x's are not part of the exponent**

thanks in advance.

Answer 1

y(sqrt(x)) - x(sqrt(y)) = 16

There's no other choice but to use implicit differentiation. Note that the derivative of sqrt(x) is 1/[2sqrt(x)]. Armed with this knowledge, we differentiation implicitly with respect to x. We use the product rule and the chain rule.

(dy/dx) [sqrt(x)] + y(1/[2sqrt(x)]) - [sqrt(y) + x(1/[2sqrt(y)])(dy/dx)] = 0

Now, let's distribute that minus sign.

(dy/dx) [sqrt(x)] + y(1/[2sqrt(x)]) - sqrt(y) - x(1/[2sqrt(y)])(dy/dx) = 0

Let's keep everything with a (dy/dx) on the left hand side; move everything else to the right hand side.

(dy/dx) (sqrt(x)) - x(1/[2sqrt(y)])(dy/dx) = sqrt(y) - y(1/[2sqrt(x)])

Factor out (dy/dx), to obtain

(dy/dx) [sqrt(x) - x(1/[2sqrt(y)])] = sqrt(y) - y(1/[2sqrt(x)])

Simplifying a bit more,

(dy/dx) [sqrt(x) - (x/[2sqrt(y)])] = sqrt(y) - (y/[2sqrt(x)])

Now, we can isolate dy/dx by dividing by the appropriate term.

dy/dx = [sqrt(y) - (y/[2sqrt(x)])] / [sqrt(x) - (x/[2sqrt(y)])]

The problem we have is that this is a complex fraction. We have to make it into a simple fraction by multiplying top and bottom by
2sqrt(x)sqrt(y). This gives us

dy/dx = [2y(sqrt(x)) - y(sqrt(y))] / [2x(sqrt(y)) - x(sqrt(x))]

On the top, we can factor out a y, and on the bottom, we can factor out an x. Making our answer slightly cleaner,

dy/dx = (y/x) [2sqrt(x) - sqrt(y)] / [2sqrt(y) - sqrt(x)]

We *could* make efforts to rationalize the denominator, but I'll stop here.