How do you find the Tangent line to a curve by implicit differentiation?

Question

Stanley Mcfarland · Accepted Answer

Find the equation of the tangent line to the circle

x^{2} + y^{2} = 5^{2}

at the point (3,4).
In order to identify a line, we need two pieces of information:

{\begin{cases} P o i n t : (x_{1}, y_{1}) = (3, 4) \\ S l o p e : m = ? \end{cases}

Since the point is already provided, all you need is the slope m.
Let us find m by implicit differentiation.
By implicitly differentiating,

\frac{d}{d x} (x^{2} + y^{2}) = \frac{d}{d x} (5^{2}) \Rightarrow 2 x + 2 y \frac{d y}{d x} = 0

by dividing by 2y,

\frac{x}{y} + \frac{d y}{d x} = 0

by subtracting

\frac{x}{y}

,

\frac{d y}{d x} = - \frac{x}{y}

So, we can find m by evaluating

\frac{d y}{d x}

at (3,4).

m = \frac{d y}{d x} ∣_{(3, 4)} = - \frac{3}{4}

By Point-Slope Form:

y - y_{1} = m (x - x_{1})

Tangent Line:

y - 4 = - \frac{3}{4} (x - 3)

How do you find the Tangent line to a curve