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
 
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}}

 
 provided this limit exists. A function is differentiable at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}
 when its 
 derivative exists at . The process of finding derivatives is called differentiation.