Implicit differentiationI’

Question

A) consider the curve given by the equation (sqrt x) + (sqrt y) =3

find the derivative y’ .

y’= -((y^(1/2))/(x^(1/2)))

B) use your answer from an and the equation of the curve to find y’(x). That is y’ as a fiction of y only.

y’(x)=?

C) Evaluate y’(1)=?

D) Use your answer from b to find y’’(x) as a function of x only.

y’’(x)=?

Answer 1

Hi Crystal!

A) In exponential notation, the relation is x^1/2 + y^1/2 = 3.

Differentiate the given relation implicitly, term-by-term, with respect to x:

(x^1/2)^' + (y^1/2)^' = 3^' (primes denote differentiation with respect to x)

1/2 x^-1/2 + 1/2 y^-1/2 y^' = 0

1/(2√x) + y^'/(2√y) = 0

Algebraically solve for y^':

y^'/(2√y) = -1/(2√x) <----- Multiply both sides by 2√y.

y^' = -√y/√x

B) Solve the original relation for y.

√x + √y = 3

√y = 3 - √x

Plug this into y^' for √y.

y^' = -(3 - √x)/√x <----- Distribute the - sign through the numerator.

y^' = (√x - 3)/√x

C) Plug x = 1 into y'.

y^'(1) = (√1 - 3)/√1

y^'(1) = -2

D) Simplify y^' to make it easier to differentiate.

y^' = ((√x - 3)/√x)

y^' = √x/√x - 3/√x

y^' = 1 - 3x^-1/2

Differentiate y^' with respect to x.

y^'' = (3/2)x^-3/2

y^'' = 3/(2√x³)

y^'' = 3/(2x√x)

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