009A Sample Midterm 1, Problem 3

Let $y={\sqrt {3x-5}}.$

a) Use the definition of the derivative to compute ${\frac {dy}{dx}}$ for $y={\sqrt {3x-5}}.$

b) Find the equation of the tangent line to $y={\sqrt {3x-5}}$ at $(2,1).$

Foundations:
1. Limit Definition of Derivative
$f'(x)=\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}$
2. Equation of a tangent line
The equation of the tangent line to $f(x)$ at the point $(a,b)$ is
$y=m(x-a)+b$ where $m=f'(a).$

Solution:

(a)

Step 1:
Let $f(x)={\sqrt {3x-5}}.$
Using the limit definition of the derivative, we have
${\begin{array}{rcl}\displaystyle {f'(x)}&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {{\sqrt {3(x+h)-5}}-{\sqrt {3x-5}}}{h}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {{\sqrt {3x+3h-5}}-{\sqrt {3x-5}}}{h}}.}\end{array}}$

Step 2:
Now, we multiply the numerator and denominator by the conjugate of the numerator.
Hence, we have
${\begin{array}{rcl}\displaystyle {f'(x)}&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {({\sqrt {3x+3h-5}}-{\sqrt {3x-5}})}{h}}{\frac {({\sqrt {3x+3h-5}}+{\sqrt {3x-5}})}{({\sqrt {3x+3h-5}}+{\sqrt {3x-5}})}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {(3x+3h-5)-(3x-5)}{h({\sqrt {3x+3h-5}}+{\sqrt {3x-5}})}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {3h}{h({\sqrt {3x+3h-5}}+{\sqrt {3x-5}})}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {3}{{\sqrt {3x+3h-5}}+{\sqrt {3x-5}}}}}\\&&\\&=&\displaystyle {\frac {3}{{\sqrt {3x-5}}+{\sqrt {3x-5}}}}\\&&\\&=&\displaystyle {{\frac {3}{2{\sqrt {3x-5}}}}.}\end{array}}$

(b)

Step 1:
We start by finding the slope of the tangent line to $f(x)={\sqrt {3x-5}}$ at $(2,1).$
Using the derivative calculated in part (a), the slope is
${\begin{array}{rcl}\displaystyle {m}&=&\displaystyle {f'(2)}\\&&\\&=&\displaystyle {\frac {3}{2{\sqrt {3(2)-5}}}}\\&&\\&=&\displaystyle {{\frac {3}{2}}.}\end{array}}$

Step 2:
Now, the tangent line to $f(x)={\sqrt {3x-5}}$ at $(2,1)$
has slope $m={\frac {3}{2}}$ and passes through the point $(2,1).$
Hence, the equation of this line is
$y={\frac {3}{2}}(x-2)+1.$

Final Answer:
(a)     ${\frac {3}{2{\sqrt {3x-5}}}}$
(b)     $y={\frac {3}{2}}(x-2)+1$

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