Use the definition of the derivative to compute d y d x {\displaystyle {\frac {dy}{dx}}} for y = 3 − 2 x + 5 . {\displaystyle y=3{\sqrt {-2x+5}}.}
Solution:
f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h = lim h → 0 3 − 2 ( x + h ) + 5 − 3 − 2 x + 5 h = lim h → 0 3 − 2 x + − 2 h + 5 − 3 − 2 x + 5 h . {\displaystyle {\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}}}
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