Find dy/dx by implicit differentiation. √x+y +√xy=4

Question

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Find dy/dx by implicit differentiation. √x+y +√xy=4

Answer 1

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Answer 2

√(x+y) +√(xy)=4

(1/2)(x+y)^(-1/2)(1+dy/dx)+(1/2)(xy)^(-1/2)(x(dy/dx)+y)=0

-(x+y)^(-1/2)-((1/2)(xy)^(-1/2))y=(dy/dx)[(1/2)(x+y)^(-1/2)+(1/2)(xy)^(-1/2)x

dy/dx=[-(x+y)^(-1/2)-(xy)^(-1/2)]y/[(x+y)^(-1/2)+((xy)^(-1/2))x]

dy/dx=-[(xy)^(1/2)(x+y)^(-1/2)-y]/[((xy)^(1/2)(x+y)^(-1/2)+x)

dy/dx=[-(xy)^(1/2)-y(x+y)^(1/2)]/[(xy)^(1/2)+x(x+y)^(1/2)]

to find dy/dx for any x,,,must solve √x+y +√xy=4 for y given an x value

and use both x and y to compute dy/dx

example. At x=1, ,,,y =225/64 according to expression given.

dy/dx at x=1 is;

y'(1)=[-(15/8)-(225/64)(17/8)]/(15/8)+(17/8)]

y'(1)=(-)2.34

Answer 3

for implicit differentiation question remember the following:

pure x function still follow the same rule -> d/dx (x) = x

pure y function still kinda follow the same rule -> d/dx (y) = dy/dx

what you need is take derivative for left and right and isolate all the dy/dx term.

step 1.

d/dx (√x+y +√xy) = d/dx(4)

step 2. right side is zero(derivative of a constant is 0). left side could be break down to simpler form using sum rule and product rule. take a look at this: https://www.mathsisfun.com/calculus/derivatives-rules.html if you forget what those are.

d/dx(√x)+d/dx(y) + √x* d/dx(y) +d/dx(√x)*y = 0

step3. expand term. we are almost there.

(1/2)*x^(-1/2) +dy/dx + √x dy/dx + (1/2)*x^(-1/2)*y = 0

step 4. move non dy/dx term to the right.

dy/dx + √x dy/dx = -(1/2)*x^(-1/2) - (1/2)*x^(-1/2)*y

step 5 grouping and simplify

(1+√x) * dy/dx = (-(1/2)*x^(-1/2)) *(-1+y)

step 6 solve for dy/dx

dy/dx = ((-(1/2)*x^(-1/2)) *(-1+y))/(1+√x)