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

Question

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

1 Answer

Impact of this question

10512 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2014-09-20T11:15:36

Let us this example:

Find the equation of the tangent line to the circle $x^{2} + y^{2} = 5^{2}$ $x^2+y^2=5^2$ at the point $(3, 4)$ $(3,4)$ . In order to identify a line, we need two pieces of information:
${("Point: " (x_1,y_1)=(3,4)), ("Slope: " m=?):}$

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

By implicitly differentiating,

$d/{dx}(x^2+y^2)=d/{dx}(5^2)Rightarrow 2x+2y{dy}/{dx}=0$

by dividing by $2y$ ,

${x}/{y}+{dy}/{dx}=0$

by subtracting $x/y$ ,

${dy}/{dx}=-x/y$

So, we can find $m$ by evaluating ${dy}/{dx}$ at $(3,4)$ .

$m={dy}/{dx}|_{(3,4)}=-3/4$

By Point-Slope Form: $y-y_1=m(x-x_1)$ ,

Tangent Line: $y-4=-3/4(x-3)$

Science

Math

Humanities

... and beyond

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

1 Answer

Related questions

Impact of this question