Math 22 The Derivative and the Slope of a Graph

Slope of a Graph

We can estimate the slope at the given point to be

Slope = ${\frac {\Delta y}{\Delta x}}={\frac {\text{change in y}}{\text{change in x}}}$

Difference Quotient

 The slope $m$ of the graph of $f$ at the point $(x,f(x))$ can be 
 written as :
 
 $m=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}$
 
 The right side of this equation ${\frac {f(x+h)-f(x)}{h}}$ is called Difference Quotient

Example: Find the Different Quotient of

1) $f(x)=x^{2}-1$

Solution: Consider ${\frac {f(x+h)-f(x)}{h}}={\frac {(x+h)^{2}-1-(x^{2}-1)}{h}}$

$={\frac {x^{2}+2xh+h^{2}-1-x^{2}+1}{h}}={\frac {2xh+h^{2}}{h}}={\frac {h(2x+h)}{h}}=2xh$

2) $f(x)=4x+1$

Solution:
Consider ${\frac {f(x+h)-f(x)}{h}}={\frac {4(x+h)+1-(4x+1)}{h}}={\frac {4x+4h+1+4x-1}{h}}={\frac {4h}{h}}=4$

Definition of the Derivattive

 The derivative of $f$ at $x$ is given by
 
 $f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}$
 
 provided this limit exists. A function is differentiable at $x$ when its 
 derivative exists at . The process of finding derivatives is called differentiation.

Example: Use limit definition to find the derivative of

1) $f(x)=x^{2}+2x$

Solution: Consider: $f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}=\lim _{h\to 0}{\frac {(x+h)^{2}+2(x+h)-(x^{2}+2x)}{h}}$

$=\lim _{h\to 0}{\frac {x^{2}+2xh+h^{2}+2x+2h-x^{2}-2x}{h}}=\lim _{h\to 0}{\frac {2xh+h^{2}+2h}{h}}$

$\lim _{h\to 0}{\frac {h(2x+h+2)}{h}}=\lim _{h\to 0}(2x+h+2)=2x+(0)+2=2x+2$

2) $f(x)=2x^{2}-3x+1$

Solution:
Consider: $f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}=\lim _{h\to 0}{\frac {2(x+h)^{2}-3(x+h)+1-(x^{2}-3x+1)}{h}}$
$=\lim _{h\to 0}{\frac {2x^{2}+4xh+2h^{2}-3x-3h+1-2x^{2}+3x-1}{h}}=\lim _{h\to 0}{\frac {4xh+2h^{2}-3h}{h}}$
$\lim _{h\to 0}{\frac {h(4x+2h-3)}{h}}=\lim _{h\to 0}(4x+2h-3)=4x+2(0)-3=4x-2$

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