009A Sample Midterm 3, Problem 3

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

Foundations:
Limit Definition of Derivative

Solution:

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

Step 2:

Final Answer:

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