The Tangent and Velocity

Tangent -Latin word “Tangens” means Touching. When a curve y = f(x) has a tangent at a point, slope of the tangent at the point is equal to the slope of the curve at the point here

Slope of Tangent line 

Slope\space of\space  line\space AB =\frac {f(c+Δx)-f(c)} {(c+Δx)-c}=\frac{f\left(c+Δx\right)-f\left(c\right)}{Δx}

Derivative of a function using limit (first principle)

Slope\space of \space tangent\space at\space A(c,\space f(c))=\lim _{Δ\:x\to 0}\frac{f\left(c+Δx\right)-f\left(c\right)}{Δx} \space or \space=\lim _{h\to 0}\frac{f\left(c+h\right)-f\left(c\right)}{h}

Slope of the Tangent line is limit of slopes of secant lines

Properties of Limits

1.\space\space\lim _{x\to \:a}\left(cf\left(x\right)\right)=c\:\lim _{x\to \:a}\left(f\left(x\right)\right)
2.\space\space\lim _{x\to \:a}\left(f\left(x\right)±g\left(x\right)\right)=\:\lim _{x\to \:a}f\left(x\right)±\:\lim _{x\to \:\:a}g\left(x\right)
3.\space\space\lim _{x\to \:a}\left(f\left(x\right).g\left(x\right)\right)=\:\lim _{x\to \:a}f\left(x\right).\:\lim _{x\to \:\:a}g\left(x\right)
4.\space\space\lim _{x\to \:a}\frac{f\left(x\right)}{g\left(x\right)}=\frac{\lim _{x\to \:a}\left(f\left(x\right)\right)}{\lim _{x\to\:a }\left(g\left(x\right)\right)}\space where \space \:\lim _{x\to \:\:a}g\left(x\right)≠0
5.\space\space\lim _{x\to \:a}\left(f\left(x\right)\right)^n=\left[\lim _{x\to \:a}\left(f\left(x\right)\right)\right]^n
6.\space\space\lim _{x\to \:a}\left(\sqrt[n]{f\left(x\right)}\right)=\sqrt[n]{\lim _{x\to \:a}\left(f\left(x\right)\right)}
7.\space\space\lim _{x\to a}\left(c\right)=c,\space where\space c\space is\space a \space real \space number \space(a\space constant)
8.\space\space\lim _{x\to a}\left(x\right)=a
9.\space\space\lim _{x\to a}\left(x^n\right)=a^n