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Find an equation of the tangent line to the curve at the given point. y = x , (81, 9)

The aim of this question is to deduce the equation of a tangent line of a curve at any point on the curve.

For any given function $ y = f(x) $, the equation of its tangent line is defined by the following equation:

\[ \boldsymbol{ y – y_1 = \frac{ dy }{ dx } ( x – x_1 ) } \]

Here, $ ( x_1 , y_1 ) $ is the point on the curve $ y = f(x) $ where the tangent line is to be evaluated and $ \dfrac{ dy }{ dx } $ is the value of the derivative of the  subject curve evaluated at the required point.

Expert Answer

Given that:

\[ y = \sqrt{ x } \]

Calculating the derivative of $y$ with respect to $x$:

\[ \frac{ dy }{ dx } = \frac{ 1 }{ 2 \sqrt{ x } } \]

Evaluating above derivative at given point $( 81 , 9 )$:

\[ \frac{ dy }{ dx } |_{ ( 81 , 9 ) } = \frac{ 1 }{ 2 \sqrt{ 81 } } \]

\[ \frac{ dy }{ dx } |_{ ( 81 , 9 ) } = \frac{ 1 }{ 2 ( 9 ) } \]

\[ \frac{ dy }{ dx } |_{ ( 81 , 9 ) } = \frac{ 1 }{ 18 } \]

The equation of a tangent line with slope $\dfrac{ dy }{ dx }$ and point $( x_1 , y_1 )$ is defined as:

\[ y – y_1 = \frac{ dy }{ dx } ( x – x_1 ) \]

Substituting values of $ \dfrac{ dy }{ dx } = \dfrac{ 1 }{ 18 } $ and point $( x_1 , y_1 ) = ( 81 , 9 ) $ in above equation:

\[ y – 9 = \frac{ 1 }{ 18 } ( x – 81 ) \]

\[ y – 9 = \frac{ 1 }{ 18 } x – \frac{ 1 }{ 18 } 81 \]

\[ y – 9 = \frac{ 1 }{ 18 } x – \frac{ 9 }{ 2 } \]

\[ y = \frac{ 1 }{ 18 } x – \frac{ 9 }{ 2 } + 9 \]

\[ y = \frac{ 1 }{ 18 } x + \frac{ – 9 + ( 2 ) ( 9 ) }{ 2 } \]

\[ y = \frac{ 1 }{ 18 } x + \frac{ – 9 + 18 }{ 2 } \]

\[ \boldsymbol{ y = \frac{ 1 }{ 18 } x + \frac{ 9 }{ 2 } }\]

Numerical Result

\[ \boldsymbol{ y = \frac{ 1 }{ 18 } x + \frac{ 9 }{ 2 } }\]

Example

Find an equation of the tangent line to the curve $y = x$ at $(1, 10)$.

Here:

\[ \frac{ dy }{ dx } = 1 \]

Using the tangent equation with $ \dfrac{ dy }{ dx } = 1 $ and point $( x_1 , y_1 ) = ( 1 , 10 ) $:

\[ y – y_1 = \frac{ dy }{ dx } ( x – x_1 ) \]

\[ y – 10 = ( 1 ) ( x – 1 ) \]

\[ y = ( 1 ) ( x – 1 ) + 10 = x – 1 + 10 \]

\[ \boldsymbol{ y = x + 9 } \]

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