To determine [tex]\(\frac{dy}{dx}\)[/tex] by implicit differentiation for the equation [tex]\(x^2 + y^2 = 1\)[/tex], follow these steps:
1. Start with the given implicit function:
[tex]\[
x^2 + y^2 = 1
\][/tex]
2. Differentiate both sides with respect to [tex]\(x\)[/tex]:
When differentiating implicitly, remember to apply the chain rule to terms involving [tex]\(y\)[/tex], since [tex]\(y\)[/tex] is a function of [tex]\(x\)[/tex].
[tex]\[
\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(1)
\][/tex]
This breaks down as follows:
[tex]\[
\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = 0
\][/tex]
The right side is a constant, so its derivative is 0. Now differentiate the left side term by term.
3. Differentiate [tex]\(x^2\)[/tex]:
[tex]\[
\frac{d}{dx}(x^2) = 2x
\][/tex]
4. Differentiate [tex]\(y^2\)[/tex]:
Using the chain rule for [tex]\(y^2\)[/tex]:
[tex]\[
\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}
\][/tex]
Combine these results:
[tex]\[
2x + 2y \frac{dy}{dx} = 0
\][/tex]
5. Solve for [tex]\(\frac{dy}{dx}\)[/tex]:
Isolate [tex]\(\frac{dy}{dx}\)[/tex]:
[tex]\[
2y \frac{dy}{dx} = -2x
\][/tex]
Simplify:
[tex]\[
y \frac{dy}{dx} = -x
\][/tex]
Finally, solve for [tex]\(\frac{dy}{dx}\)[/tex]:
[tex]\[
\frac{dy}{dx} = -\frac{x}{y}
\][/tex]
Thus,
[tex]\[
\boxed{ \frac{dy}{dx} = -\frac{x}{y} }
\][/tex]
