Find the derivative by solving the given implicit equation for y explicitly in terms of x.
C) 2x
D) 2

Question

Find the derivative  by solving the given implicit equation for y explicitly in terms of x. ​  ​
C) 2x 
D) 2

Quizplus · Accepted Answer

We have the implicit equation:

$11eb651f_cb62_989a_b0a1_63daf280fcee_TB8255_11$,

We need to solve this for $y$ explicitly in terms of $x$.

Rewriting the equation:

$(y^2 - 4)^2 = 4(x^2 - 1)$

Taking the square root of both sides:

$y^2 - 4 = \pm 2\sqrt{x^2 - 1}$

Adding 4 to both sides:

$y^2 = 4 \pm 2\sqrt{x^2 - 1}$

Taking the square root of both sides (remembering that we need to include both the positive and negative square roots):

$y = \pm \sqrt{4 \pm 2\sqrt{x^2-1}}$

Now we can take the derivative of both sides with respect to $x$. Using the chain rule on the right-hand side:

$\frac{dy}{dx} = \pm \frac{1}{2\sqrt{4 \pm 2\sqrt{x^2-1}}} \cdot \frac{d}{dx}\left(4 \pm 2\sqrt{x^2-1}ight)$

$= \pm \frac{1}{2\sqrt{4 \pm 2\sqrt{x^2-1}}} \cdot \left(0 \pm \frac{2x}{\sqrt{x^2-1}}ight)$

$= \pm \frac{x}{\sqrt{(4 \pm 2\sqrt{x^2-1})(x^2-1)}}$

So the final answer is:

$\boxed{	extbf{(B)}\ \frac{x}{\sqrt{(4 \pm 2\sqrt{x^2-1})(x^2-1)}}}$

Find the Derivative by Solving the Given Implicit Equation