CCD.DDN43: There are four steps to calculating the derivative of a function $f$ based primarily on analysis of the difference quotient used in the definition . 

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Alternative $\Delta x = x-a$:
$f '(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}$
Step I: Evaluate $f (x)$ and $f (a)$.
Step II: Subtract: Find (and simplify when possible) the difference:   $\Delta y = f (x) - f (a)$.
Step III: Divide by $\Delta x = x - a$  to find $\frac{f(x)-f(a)}{x-a}$ and then simplify algebraically (when possible) to eliminate $\Delta x = x - a$ from the denominator.
Step IV: Think! Finally, analyze the simplified expression to see what happens when $x\to a$, remembering that $x \ne a$.
If the last expression, $\frac{\Delta y}{\Delta x}=\frac{f(x)-f(a)}{x-a}$ approaches a single number,$L$, then $L=f '(a)$.

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Alternative $\Delta x= h$. The four steps can use $x=a+h$ and $\Delta x = h$.
$\ f'(a) = \lim_{h \to 0}\frac {f(a+h) - f(a)}{h}$
Step I: Evaluate: $f (a+h)$ and $f (a)$.
Step II: Subtract: Find (and simplify when possible) the difference:   $\Delta y = f (a+h) - f (a)$.
Step III: Divide by $\Delta x = h$ to find $\frac{f(a+h)-f(a)}{h}$ and then simplify algebraically (when possible) to eliminate $\Delta x = h$ from the denominator.
Step IV: Think! Finally, analyze the simplified expression to see what happens when $h\to 0$, remembering that $h \ne 0$.
If the last expression, $\frac{f(a+h)-f(a)}{h}$ approaches a single number,$L$, then $L=f '(a)$.

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Alternative $\Delta x$:
$\ f'(a) = \lim_{\Delta x \to 0}\frac {f(a+\Delta x) - f(a)}{\Delta x}$
Step I. Evaluate: $f(a+\Delta x)$ and $f(a)$
Step II. Subtract:  Find (and simplify when possible) the difference: $\Delta y =f(a+\Delta x) - f(a)$
Step III. Divide: $\frac {\Delta y}{\Delta x} =\frac {f(a+\Delta x) - f(a)}{\Delta x}$ and simplify if possible.
Step IV. Think! As $\Delta x \to 0$, does $\frac {f(a+\Delta x) - f(a)}{\Delta x} \to L$ ? If so, then $L =f'(a)$.




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