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Differentiation Fundamentals

Calculus is a branch of mathematics that studies motion and how things change. 

A secant line to a curve is a line that intersects the curve at least at two distinct points. The slope of the secant line at the points \(P(a,f(a))\) and \(Q(b,f(b))\) is found computing the difference quotient

\(\displaystyle m_{secant}=\frac{f(b)-f(a)}{b-a}\)

A secant line can be used to approximate the tangent line to a curve. 

Tangent Line

The derivative of a function \(f(x)\), which is synonymous with the instantaneous rate of change (or simply rate of change), is defined as follows 

\(\displaystyle\underset{h\to 0}{\lim}\frac{f(x+h)-f(x)}{h}=f'(x)\)


The derivative of a function \(f(x)\) at a given value \(x=a\) is defined as follows

\(\displaystyle\underset{b\to a}{\lim}\frac{f(b)-f(a)}{b-a}=f'(a)\)

The difference quotient for a function \(f(x)\) is defined as the function 

\(F(x)=\displaystyle\frac{f(x+h)-f(x)}{h}\)

As you perform the following steps simplify as you go. 

Compute the composition: \(\displaystyle f(x+h)=\)
Compute the difference: \(\displaystyle f(x+h)-f(x)=\)
Compute the quoitent:\(\displaystyle\frac{f(x+h)-f(x)}{h}=\)

The average rate of change measures how a function \(f(x)\) is changing on from a point \(P\big((a,f(a)\big)\) to point \(Q\big(b,f(b)\big)\)

\(m=\displaystyle\frac{f(b)-f(a)}{b-a}\)

which may be expressed as follows where \(b=a+h\)

\(m=\displaystyle\frac{f(a+h)-f(a)}{a}\)

The average rate of change corresponds
to the slope of a secant line.

The instantaneous rate of change measures how a function \(f(x)\) is changing at a moment’s notice at a given point \(P\big((a,f(a)\big)\). 

\(f'(a)=\displaystyle \underset{h\to 0}{\lim}\frac{f(a+h)-f(a)}{h} =\underset{b\to a}{\lim}\frac{f(b)-f(a)}{b-a}\)

The instantaneous rate of change
corresponds to the slope of a tangent line.

Definitions to Know

Invalidies Differentiability

Differential functions are ALWAYS continuous functions, but continuous may not be differentiable. Functions that are not differentiable at \(x=a\) means that the slope of the tangent line is undefined at the point \((a,f(x))\). This phenomenon can occur at sharp corners, cusps, discontinuities, or vertical tangent lines. [A cusp occurs when the two different one-sided limits in the limit definition of a derivative yield different values.]

Sharp Corners

Cusps

Discontinuity

Vertical Tangent Lines

\(\displaystyle\frac{f(x+h)-f(x)}{h}\)

The average rate of change measures how a function \(f(x)\) is changing on from a point \(P\big((a,f(a)\big)\) to point \(Q\big(b,f(b)\big)\)

\(m=\displaystyle\frac{f(b)-f(a)}{b-a}\)

which may be expressed as follows where \(b=a+h\)

\(m=\displaystyle\frac{f(a+h)-f(a)}{a}\)

The instantaneous rate of change measures how a function \(f(x)\) is changing at a moment’s notice at a given point \(P\big((a,f(a)\big)\). 

\(\displaystyle \underset{h\to 0}{\lim}\frac{f(a+h)-f(a)}{h}\)

Secant Line
Tangent Line

The derivative of a function \(f(x)\), which is synonymous with the instantaneous rate of change (or simply rate of change), is defined as follows 

\(\displaystyle\underset{h\to 0}{\lim}\frac{f(x+h)-f(x)}{h}=f'(x)\)


The derivative of a function \(f(x)\) at a given value \(x=a\) is defined as follows

\(\displaystyle\underset{b\to a}{\lim}\frac{f(b)-f(a)}{b-a}=f'(a)\)