(1) Slope of
the tangent : If a tangent is drawn to the curve \[y=f(x)\] at a point
\[P({{x}_{1}},\,{{y}_{1}})\] and this tangent makes an angle \[\psi \] with positive x-direction then, \[{{\left(
\frac{dy}{dx} \right)}_{({{x}_{1}},\,{{y}_{1}})}}=\tan \psi =\] Slope of the
tangent.
·
If the tangent is parallel to x-axis, \[\psi =0\Rightarrow
{{\left( \frac{dy}{dx} \right)}_{({{x}_{1}},\,{{y}_{1}})}}=0\]
·
If the tangent is perpendicular to x-axis, \[\psi
=\frac{\pi }{2}\Rightarrow {{\left( \frac{dy}{dx}
\right)}_{({{x}_{1}},\,{{y}_{1}})}}\to \,\,\,\infty \]
(2) Slope of the normal : The normal to a
curve at a point \[P({{x}_{1}},\,{{y}_{1}})\] is a line perpendicular to the tangent at P and passing
through P. Slope of the normal \[=\frac{-1}{\text{Slope of
tangent }}=\frac{-1}{{{\left( \frac{dy}{dx}
\right)}_{P({{x}_{1}},\,{{y}_{1}})}}}=-{{\left( \frac{dx}{dy}
\right)}_{P({{x}_{1}},\,{{y}_{1}})}}\].
·
If the normal is parallel to x-axis, \[-{{\left(
\frac{dx}{dy} \right)}_{({{x}_{1}},\,{{y}_{1}})}}=0\] or
\[\frac{b}{a}=\frac{c}{b}\].
·
If the normal is perpendicular to x-axis
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