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EXAMPLE 1 (a) If $x^2 + y^2 = 225$, find $\frac{dy}{dx}$. (b) Find an equation of the tangent line to the circle $x^2 + y^2 = 225$ at the point $(-9, -12)$. SOLUTION (a) Differentiating both sides of the equation $x^2 + y^2 = 225$: $\frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}225$ $\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = 0$ Remembering that $y$ is a function of $x$ and using the Chain Rule, we have $\frac{d}{dx}y^2 = \frac{d}{dy}y^2 \frac{dy}{dx} = 2y\frac{dy}{dx}$ Thus $2x + 2y\frac{dy}{dx} = 0$ Now we solve this equation for $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{-x}{y}$ (b) An equation of the tangent to the circle at $(-9, -12)$ is therefore $y - (-12) = -\frac{3}{4}(x - (-9))$ or $-\frac{3}{4}x - \frac{75}{4} = y = 225$

          EXAMPLE 1 (a) If $x^2 + y^2 = 225$, find $\frac{dy}{dx}$.
(b) Find an equation of the tangent line to the circle $x^2 + y^2 = 225$ at the point $(-9, -12)$.
SOLUTION (a) Differentiating both sides of the equation $x^2 + y^2 = 225$:
$\frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}225$
$\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = 0$
Remembering that $y$ is a function of $x$ and using the Chain Rule, we have
$\frac{d}{dx}y^2 = \frac{d}{dy}y^2 \frac{dy}{dx} = 2y\frac{dy}{dx}$
Thus
$2x + 2y\frac{dy}{dx} = 0$
Now we solve this equation for $\frac{dy}{dx}$:
$\frac{dy}{dx} = \frac{-x}{y}$
(b) An equation of the tangent to the circle at $(-9, -12)$ is therefore
$y - (-12) = -\frac{3}{4}(x - (-9))$
or $-\frac{3}{4}x - \frac{75}{4} = y = 225$

EXAMPLE 1 (a) If x^2 + y^2 = 225, find (dy)/(dx).
(b) Find an equation of the tangent line to the circle x^2 + y^2 = 225 at the point (-9, -12).
SOLUTION (a) Differentiating both sides of the equation x^2 + y^2 = 225:
(d)/(dx)(x^2 + y^2) = (d)/(dx)225
(d)/(dx)(x^2) + (d)/(dx)(y^2) = 0
Remembering that y is a function of x and using the Chain Rule, we have
(d)/(dx)y^2 = (d)/(dy)y^2 (dy)/(dx) = 2y(dy)/(dx)
Thus
2x + 2y(dy)/(dx) = 0
Now we solve this equation for (dy)/(dx):
(dy)/(dx) = (-x)/(y)
(b) An equation of the tangent to the circle at (-9, -12) is therefore
y - (-12) = -(3)/(4)(x - (-9))
or -(3)/(4)x - (75)/(4) = y = 225

Added by Michael D.

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