Example of the Slope of a Tangent

In this tutorial we shall discuss an example of the slope of a tangent to any curve at some given point.

Consider the slope of a tangent to {x^2} = 16y at the point \left( {4,1} \right)

We have

{x^2} = 16y\,\,\,\,\,{\text{ - - - }}\left( {\text{i}} \right)

Differentiating both sides with respect to x, we get

\begin{gathered} \frac{d}{{dx}}{x^2} = 16\frac{d}{{dx}}y \\ \Rightarrow 2x = 16\frac{{dy}}{{dx}} \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{x}{8}\,\,\,\,\,\,{\text{ - - - }}\left( {{\text{ii}}} \right) \\ \end{gathered}

For the slope of the tangent at the given point \left( {4,1} \right), put x = 4,\,y = 1 in equation (ii) to get

{\frac{{dy}}{{dx}}_{\left( {4,1} \right)}} = \frac{4}{8} = \frac{1}{2}

Thus, the slope of the tangent to {x^2} = 16y at the point \left( {4,1} \right) is \frac{1}{2}.