Derivatives

Summary

If \(a\) and \(b\) with \(b>a\) are two points in the domain of a function \(f\) such that \((a,f(a))\) and \((b,f(b))\) are two points on the graph then the straight line through these points is called a secant .
The slope of the secant through \((a,f(a))\) and \((b,f(b))\) is given by

\[ \frac{f(b)-f(a)}{b-a}\]

If you define \(x_0=a\) and \(h=b-a\) then the slope of the secant through \((x_0,f(x_0))\) and \((x_0+h,f(x_0+h))\) is given by

\[ \frac{f(x_0+h)-f(x_0)}{h}\]

\[\lim_{h→0} \frac{f(x_0+h)-f(x_0)}{h}\]

exists then we say that \(f\) is differentiable at \(x=x_0\) and we denote the limit by \(f'(x_0)\) called the derivative of \(f\) at \(x=x_0\) . The derivative at \(x=x_0\) can equivalently be defined as

\[\lim_{x→x_0} \frac{f(x)-f(x_0)}{x-x_0}\]

If \(f\) is differentiable at \(x=x_0\) then the straight line through \((x_0,f(x_0))\) with slope \(f'(x_0)\) is called the tangent of \(f\) at \(x=x_0\) . The tangent has equation

\[T_1(x)=f(x_0)+f'(x_0)(x-x_0)\]

If \(f:A→B\) we say that \(f\) is differentiable on \(C⊆A\) if \(f\) is differentiable at every point of \(C\) .
If \(f\) is differentiable on \(C\) , then \(f':C→R\) is a function on \(C\) called the derivative of \(f\) on \(C\) .
We typically use the same name for the input variable for \(f\) and \(f'\) ; if \(f(x)=x^2\) we write \(f'(x)=2x\) .
Alternative notation for \(f'\left( x \right)\) :

\[ \frac{df\left( x \right)}{dx} df(x)/dx \frac{df}{dx}\left( x \right) \frac{d}{dx}f\left( x \right) \frac{dy}{dx} dy/dx y'\]

Example: we can write “if \(f\left( x \right)=x^2\) then \(f'\left( x \right)=2x\) ” or, equivalently,

\[ \frac{d}{dx}x^2=2x\]

\[ \frac{df}{dx}\]

Alternative notation for \(f'(x_0)\) , the derivative evaluated at the point \(x=x_0\) :

\[{\left. \frac{df(x)}{dx} \right|}_{x=x_0}\]