# Implicit differentiation

## Tangent lines of a circle

• Recall that the equation of the circle $$C$$ with radius 5 and centre $$O(0,0)$$ is $x^2+y^2=25.$
• Let us find equations of its tangent lines.
• One way would be to separate the circle into the two graphs $$y=\pm\sqrt{25-x^2}$$, and derivate the functions.
• Implicit differentiation gives an easier way: you can derivate the equation of the circle itself with respect to $$x$$.
• What you need to look out for is that when derivating an expression containing $$y$$, you need to derivate that as $$y(x)$$ and thus use the chain rule.
• Let's do this! We get $2x+2yy'=0.$
• Solving for $$y'$$ gives $$y'=-\frac{x}{y}$$. This is saying that the slope of the tangent line line at $$P(a,b)$$ of $$C$$ is $$y'(x=a)=-\frac{a}{b}$$.
• For example, the tangent line at $$P(3,4)$$ has equation $$y-4=-\frac34(x-3)$$.
• Exercises. 3.5: 2, 4, 10, 12, 26, 34, 46