Question

How do you find the slope of the tangent line to the graph at given point and give an equation of the tangent line \[f(x) = \dfrac{3}{2}x\] at \[\left( {1,\dfrac{3}{2}} \right)\]?

Answer 1

Hint: Here in this question given a equation of tangent line here we have to find the slope of the tangent line by differentiating the given equation of tangent line by using standard differentiation formula and further substitute the given point to the differentiate equation we get the required value of slope.

Complete step by step answer:
A tangent line is a straight line that touches the plotted curve at a single point. That point is known as the point of tangency.
The slope of a linear equation can be found with the formula: \[y = mx + b\]. When dealing with a curved line, where the slope is changing, you can’t use the same formula. You have to divide the change in y-values by the change in x-values, represented as:
\[m = \dfrac{{\begin{array}{*{20}{c}}
  {change}&{in}&y
\end{array}}}{{\begin{array}{*{20}{c}}
  {change}&{in}&x
\end{array}}}\]
Otherwise it can be written as
\[m = \dfrac{{dy}}{{dx}}\]
Now consider, the given equation of the tangent line
\[ \Rightarrow \,\,\,\,f(x) = \dfrac{3}{2}x\]
Differentiate \[f(x)\] with respect to \[x\], then
\[ \Rightarrow \,\,\,\,\dfrac{d}{{dx}}f(x) = \dfrac{d}{{dx}}\left( {\dfrac{3}{2}x} \right)\]
Slope of tangent can be represented as \[m = \dfrac{d}{{dx}}f(x) = f'(x)\]
\[ \Rightarrow \,\,\,m = \,f'(x) = \dfrac{3}{2}\dfrac{d}{{dx}}\left( x \right)\]
\[ \Rightarrow \,\,\,\,m = f'(x) = \dfrac{3}{2} \cdot \,1\]
\[ \Rightarrow \,\,\,\,m = f'(x) = \dfrac{3}{2}\]
Now the slope of the tangent line to the graph at a given point \[\left( {1,\dfrac{3}{2}} \right)\]. Substitute the point value x and y to the slope of tangent m
\[\therefore \,\,\,\,m = \dfrac{3}{2}\]
Where slope of tangent is constant so need to substitute the point value.

Hence, the slope of the tangent line to the graph at given point and give an equation of the tangent line \[f(x) = \dfrac{3}{2}x\] at \[\left( {1,\dfrac{3}{2}} \right)\] is \[m = \dfrac{3}{2}\].

Note: This concept belongs to the graph. A tangent line is a straight line which touches the plotted curve at a single point. the f(x) is a function of x and it can be called as y. here the value of y varies based on the value of x. The slope of a linear equation can be found with the formula is \[y = mx + b\].