The following examples show further applications of the principles just explained.

(4) Find the slope of the tangent to the curve

at the point where ${\displaystyle x=-1}$. Find the angle which this tangent makes with the curve ${\displaystyle y=2x^{2}+2}$.

The slope of the tangent is the slope of the curve at the point where they touch one another (see p.77); that is, it is the ${\displaystyle {\dfrac {dy}{dx}}}$ of the curve for that point. Here ${\displaystyle {\dfrac {dy}{dx}}=-{\dfrac {1}{2x^{2}}}}$ and for ${\displaystyle {\dfrac {dy}{dx}}=-{\dfrac {1}{2}}}$, which is the slope of the tangent and of the curve at that point. The tangent, being a straight line, has for equation ${\displaystyle y=ax+b}$, and its slope is ${\displaystyle {\dfrac {dy}{dx}}=a}$, hence ${\displaystyle a=-{\dfrac {1}{2}}}$. Also if ${\displaystyle x=-1}$, ${\displaystyle y={\dfrac {1}{2(-1)}}+3=2{\frac {1}{2}}}$; and as the tangent passes by this point, the coordinates of the point must satisfy the equation of the tangent, namely

so that ${\displaystyle 2{\frac {1}{2}}=-{\dfrac {1}{2}}\times (-1)+b}$ and ${\displaystyle b=2}$; the equation of the tangent is therefore ${\displaystyle y=-12x+2}$.

Now, when two curves meet, the intersection being a point common to both curves, its coordinates must satisfy the equation of each one of the two curves;