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 $$\frac{1}{2}$$.