Example of Slope of Tangent

In this tutorial we shall discuss an example of slope of tangent to any curve at some given point.
Consider the slope of 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 slope of 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 tangent to {x^2} = 16y at the point \left( {4,1} \right) is \frac{1}{2}.

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