009A Sample Midterm 3, Problem 3

Use the definition of the derivative to compute   ${\frac {dy}{dx}}$   for  $y=3{\sqrt {-2x+5}}.$

Foundations:
Recall
$f'(x)=\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}$

Solution:

Step 1:
Let  $f(x)=3{\sqrt {-2x+5}}.$
Using the limit definition of the derivative, we have
${\begin{array}{rcl}\displaystyle {f'(x)}&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {3{\sqrt {-2(x+h)+5}}-3{\sqrt {-2x+5}}}{h}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {3{\sqrt {-2x+-2h+5}}-3{\sqrt {-2x+5}}}{h}}}\\&&\\&=&\displaystyle {3\lim _{h\rightarrow 0}{\frac {{\sqrt {-2x+-2h+5}}-{\sqrt {-2x+5}}}{h}}.}\end{array}}$

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

Final Answer:
${\frac {dy}{dx}}=-{\frac {3}{\sqrt {-2x+5}}}$

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