To find [tex]\(\frac{dy}{dx}\)[/tex] by implicit differentiation, we will differentiate both sides of the equation with respect to [tex]\(x\)[/tex]. Let's go through the steps together:
First, consider the given equation:
[tex]\[
\sqrt{xy} = x^8 y + 60
\][/tex]
### Step 1: Differentiate both sides with respect to [tex]\(x\)[/tex]
Differentiate the left-hand side:
[tex]\[
\frac{d}{dx} \left( \sqrt{xy} \right)
\][/tex]
Using the chain rule on [tex]\(\sqrt{xy}\)[/tex]:
[tex]\[
\frac{d}{dx} \left( xy \right)^{1/2} = \frac{1}{2} (xy)^{-1/2} \cdot \frac{d}{dx} (xy)
\][/tex]
Now, apply the product rule to [tex]\(\frac{d}{dx} (xy)\)[/tex]:
[tex]\[
\frac{d}{dx} (xy) = x \frac{d y}{dx} + y
\][/tex]
Putting it together:
[tex]\[
\frac{d}{dx} (xy)^{1/2} = \frac{1}{2} (xy)^{-1/2} \left( x \frac{d y}{dx} + y \right)
\][/tex]
So, the derivative of the left-hand side is:
[tex]\[
\frac{1}{2} \frac{x y^{1/2}}{(xy)^{1/2}} \left( x \frac{d y}{dx} + y \right)
\][/tex]
Since [tex]\(\sqrt{xy} = (xy)^{1/2}\)[/tex], this simplifies to:
[tex]\[
\frac{1}{2 \sqrt{xy}} (x \frac{d y}{dx} + y)
\][/tex]
Now, differentiate the right-hand side:
[tex]\[
\frac{d}{dx} (x^8 y + 60)
\][/tex]
Using the product rule on [tex]\(x^8 y\)[/tex]:
[tex]\[
= 8x^7 y + x^8 \frac{d y}{dx}
\][/tex]
And the derivative of the constant 60 is 0.
So, the derivative of the right-hand side is:
[tex]\[
8 x^7 y + x^8 \frac{d y}{dx}
\][/tex]
### Step 2: Equate the derivatives and solve for [tex]\(\frac{dy}{dx}\)[/tex]
Equate the derivatives from both sides:
[tex]\[
\frac{1}{2 \sqrt{xy}} (x \frac{d y}{dx} + y) = 8x^7 y + x^8 \frac{d y}{dx}
\][/tex]
Multiply through by [tex]\(2 \sqrt{xy}\)[/tex] to clear the denominator:
[tex]\[
x \frac{d y}{dx} + y = 16 x^7 y \sqrt{xy} + 2 x^8 \frac{d y}{dx} \sqrt{xy}
\][/tex]
Collect all terms involving [tex]\(\frac{d y}{dx}\)[/tex] on one side:
[tex]\[
x \frac{d y}{dx} - 2 x^8 \frac{d y}{dx} \sqrt{xy} = 16 x^7 y \sqrt{xy} - y
\][/tex]
Factor out [tex]\(\frac{d y}{dx}\)[/tex]:
[tex]\[
\frac{d y}{dx} (x - 2 x^8 \sqrt{xy}) = 16 x^7 y \sqrt{xy} - y
\][/tex]
Finally, solve for [tex]\(\frac{d y}{dx}\)[/tex]:
[tex]\[
\frac{d y}{dx} = \frac{16 x^7 y \sqrt{xy} - y}{x - 2 x^8 \sqrt{xy}}
\][/tex]
This simplifies to:
[tex]\[
\frac{d y}{dx} = -8 x^7 y + \frac{\sqrt{xy}}{2 x}
\][/tex]
Thus, the derivative [tex]\(\frac{dy}{dx}\)[/tex] is:
[tex]\[
\frac{dy}{dx} = -8 x^7 y + \frac{\sqrt{xy}}{2 x}
\][/tex]
