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How do you use implicit differentiation to find an equation of the tangent line to the curve \[{x^2} + 2xy - {y^2} - x = 39\] at the given point \[\left( {5,9} \right)\]?

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Last updated date: 28th Feb 2024
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Hint: In this question, we need to find the equation of the tangent to the curve \[{x^2} + 2xy - {y^2} + x = 39\] at the point \[\left( {5,9} \right)\], this is done by differentiating with respect to x the given equation and then obtain the equation of the tangent line by substituting the points.

Complete Step By Step solution:
In this question we have given the equation as \[{x^2} + 2xy - {y^2} + x = 39\] and we need to find the equation of the tangent line to the given curve.
First we will differentiate the above equation with respect to x as,
\[
   \Rightarrow \dfrac{d}{{dx}}\left( {{x^2} + 2xy - {y^2} + x} \right) = \dfrac{d}{{dx}}\left( {39} \right) \\
   \Rightarrow 2x + 2y + 2x\dfrac{{dy}}{{dx}} - 2y\dfrac{{dy}}{{dx}} + 1 = 0 \\
   \Rightarrow \left( {2x - 2y} \right)\dfrac{{dy}}{{dx}} = - 1 - 2x - 2y \\
   \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{1 + 2x + 2y}}{{2y - 2x}} \\
 \]
As we can see in the above calculation the term \[2xy\] is differentiated by the chain rule.
The formula to determine the equation of the tangent line is given below.
\[y = {y_0} + y'\left( {{x_0}} \right)\left( {x - {x_0}} \right)\]
For the above equation to calculate the value of \[y'\left( {{x_0}} \right)\] we will substitute the initial condition in the equation as,
\[
   \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{1 + 2\left( 5 \right) + 2\left( 9 \right)}}{{2\left( 9 \right) - 2\left( 5 \right)}} \\
   \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{1 + 10 + 18}}{{18 - 10}} \\
 \]
After simplification we will get,
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{29}}{8}\]
Now we will get the equation for the tangent line obtained by substituting the value of the initial points as shown below.
\[
  y = 9 + \dfrac{{29}}{8}\left( {x - 5} \right) \\
   \Rightarrow y = 9 + \dfrac{{29}}{8}\left( {x - 5} \right) \\
 \]
After simplification we will get,
\[\therefore 29x - 8y = 73\]

Thus, the equation \[29x - 8y = 73\] is the equation of the tangent line that is tangent to the equation \[{x^2} + 2xy - {y^2} - x = 39\] at the point \[\left( {5,9} \right)\].

Note:
The process to find the derivative of the function is called differentiation, in differentiation the instantaneous rate of change of function based on the variable. In implicit differentiation each side of the equation with the two variables is differentiated, this is done by treating one of the variables as the other's function.