Answer:
Step-by-step explanation:
The question is incomplete. Find the complete question in the attached file.
a) Given the expression 9x^2 - y^2 = 1
Differentiating implicitly will give;
[tex]18x-2y\frac{dy}{dx} = 0\\[/tex]
We can then make dy/dx the subject of the formula as shown;
[tex]18x = 2y\frac{dy}{dx}\\\frac{dy}{dx} = \frac{18x}{2y} \\\frac{dy}{dx} = \frac{9x}{y} \\y' = \frac{9x}{y}[/tex]
b) In order to solve the question explicitly, we will first have to make x the subject of the formula before differentiating.
[tex]9x^{2} -y^{2} = 1\\9x^{2} = 1+ y^{2}\\y^{2} =9x^{2}- 1\\y^{2} = 9x^{2} - 1\\y = \sqrt{9x^{2}- 1 } \\[/tex]
Using chain rule to solve the equation;
let;
[tex]u = 9x^{2} -1\\ y = u^{1/2}[/tex]
du/dx = 18x
dy/du = [tex]1/2u^{-1/2}[/tex]
dy/dx = dy/du * du/dx
dy/dx = [tex]1/2u^{-1/2} * 18x[/tex]
[tex]\frac{dy}{dx} = \frac{1}{2}({9x^{2}-1 } )^{-1/2} * 18x\\\frac{dy}{dx} = 9x({9x^{2} -1 } )^{-1/2} \\\frac{dy}{dx} = 9 x(\sqrt{{\frac{1}{9x^{2} -1} } }) \\\frac{dy}{dx} = \frac{9x}{\sqrt{9x^{2}-1 } }[/tex]
c) In order to confrim that solutions to part (a) and (b) are consistent, we will substitute [tex]y = \sqrt{9x^{2} - 1 } \\[/tex] into the answer in (a) as shown;
From (a) [tex]\frac{dy}{dx} = \frac{9x}{y} \\[/tex]
[tex]y' = \frac{9x}{ \sqrt{9x^{2} - 1 } \\} } \\}[/tex]
This shows that they are consistent
