Question

9x^2-y^2=1
(a) find y' by implicit differentiation
(b) Solve the equation explicitly for y and differentiate to get y' in terms of x
(c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a)

Answer 1

9x^2 - y^2 = 1

a.) 18x - 2y dy/dx = 0
2y dy/dx = 18x
dy/dx = 9x/y

b.) y^2 = 9x^2 - 1
y = √(9x^2 - 1)
y' = 9x / √(9x^2 - 1)

c.) substituting y = √(9x^2 - 1) into solution for part a gives
dy/dx = 9x / √(9x^2 - 1)
Thus the two solutions are consistent.

Answer 2

What is differentiation?

Finding the rate of change in something with respect to something is known as differentiation. For Example, Change in y with respect to x is known as the derivative of y with respect to x.

How to solve it?

(a) $9{x}^2-{y}^2 = 1$

Differentiating equation with respect to x :

$9*2x - 2y\frac{dy}{dx} = 0\\ 18x = 2y\frac{dy}{dx}\\9\frac{x}{y} =\frac{dy}{dx}$

(b)

$9{x}^2-{y}^2 = 1\\9{x}^2-1 = {y}^2 \\\sqrt[2]{9{x}^2-1} = y$

differentiating with respect to x gives:

$\frac{1}{2\sqrt{9{x}^2-1} } *(18x) = \frac{dy}{dx}\\\frac{1}{\sqrt{9{x}^2-1} } *(9x) = \frac{dy}{dx}$

(c) $9\frac{x}{y} =\frac{dy}{dx}$

substituting value of y

$\sqrt[2]{9{x}^2-1} = y\\\\\frac{9x}{\sqrt[2]{9{x}^2-1}} = \frac{dy}{dx}$

Hence the solution is consistent.

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