Derivative of product rule or differentiation of product rule
let [katex]f(x)=u.v[/katex]
where [katex]u[/katex] and [katex]v[/katex] are function of [katex]x[/katex]
[katex]\frac d{dx}f\left(x\right)=\frac d{dx}\left[\left(u\right).\left(v\right)\right][/katex]
[katex]\boxed{\frac d{dx}f\left(x\right)=\left(u\right).\frac d{dx}\left(v\right)+\left(v\right).\frac d{dx}\left(u\right)}[/katex]
Derivative of quotient rule or differentiation of quotient rule
[katex]f\left(x\right)=\frac uv[/katex]
where [katex]u[/katex] and [katex]v[/katex] are function of [katex]x[/katex]
[katex]\frac d{dx}f\left(x\right)=\frac d{dx}\frac uv[/katex]
[katex]\boxed{\frac d{dx}f\left(x\right)=\frac{\left(v\right).{\displaystyle\frac d{dx}}\left(u\right)-\left(u\right).{\displaystyle\frac d{dx}}\left(v\right)}{\left(v\right)^2}}[/katex]
Derivative of power rule or differentiation of power rule
[katex]f\left(x\right)=x^n[/katex]
[katex]\frac d{dx}f\left(x\right)=\frac d{dx}x^n[/katex]
[katex]\boxed{\frac d{dx}f\left(x\right)=nx^{n-1}}[/katex]
Derivative of chain rule or differentiation of chain rule
Let [katex]y=f(u)[/katex] and [katex]u=g(x)[/katex]
[katex]\boxed{\frac{dy}{dx}=\frac{dy}{du}.\frac{du}{dx}}[/katex]
Derivative of trigonometric functions or Differentiation of trigonometric functions
[katex]\boxed{\frac d{dx}\left(\sin x\right)=\cos x}[/katex]
[katex]\boxed{\frac d{dx}\left(\cos x\right)=-sin x}[/katex]
[katex]\boxed{\frac d{dx}\left(\tan x\right)=sec^2 x}[/katex]
[katex]\boxed{\frac d{dx}\left(\cot x\right)=-cosec^2 x}[/katex]
[katex]\boxed{\frac d{dx}\left(\cosec x\right)=-cosec x.cot x}[/katex]
[katex]\boxed{\frac d{dx}\left(\sec x\right)=sec x.tan x}[/katex]
Derivative of inverse trigonometric functions ,Differentiation of inverse trigonometric functions
[katex]\boxed{\frac d{dx}(Sin^{-1}x)=\frac1{\sqrt{1-x^2}}}[/katex]
[katex]\boxed{\frac d{dx}(Cos^{-1}x)=\frac{-1}{\sqrt{1-x^2}}}[/katex]
[katex]\boxed{\frac d{dx}(Tan^{-1}x)=\frac{1}{{1+x^2}}}[/katex]
[katex]\boxed{\frac d{dx}(Cot^{-1}x)=\frac{-1}{{1+x^2}}}[/katex]
[katex]\boxed{\frac d{dx}(sec^{-1}x)=\frac1{\left|x\right|\sqrt{1-x^2}}}[/katex]
[katex]\boxed{\frac d{dx}(Cosec^{-1}x)=\frac{-1}{\left|x\right|\sqrt{1-x^2}}}[/katex]
Derivative of sum and difference rule, Differentiation of sum and difference rule
sum rule of Derivative
let [katex]y=u(x)+v(x)[/katex]
where [katex]u[/katex] and [katex]v[/katex] are function of [katex]x[/katex]
[katex]\boxed{\frac d{dx}\left(y\right)=\frac d{dx}\left[u\left(x\right)+v\left(x\right)\right]}[/katex]
difference rule of Derivative
let [katex]y=u(x)-v(x)[/katex]
where [katex]u[/katex] and [katex]v[/katex] are function of [katex]x[/katex]
[katex]\boxed{\frac d{dx}\left(y\right)=\frac d{dx}\left[u\left(x\right)-v\left(x\right)\right]}[/katex]
Derivative of hyperbolic functions
[katex]\boxed{Sinhx=\frac{e^x-e^{-x}}2}[/katex]
[katex]\boxed{Coshx=\frac{e^x+e^{-x}}2}[/katex]
[katex]\boxed{Tanhx=\frac{Sinhx}{Coshx}=\frac{e^x+e^{-x}}{e^x-e^{-x}}}[/katex]
[katex]\boxed{Cosechx=\frac1{Sinhx}=\frac2{e^x-e^{-x}}}[/katex]
[katex]\boxed{Sechx=\frac1{Coshx}=\frac2{e^x+e^{-x}}}[/katex]
[katex]\boxed{Cothx=\frac{Coshx}{Sinhx}=\frac{e^x+e^{-x}}{e^x-e^{-x}}}[/katex]
Logarithmic differentiation
[katex]\boxed{\frac d{dx}\ln x=\frac1x}[/katex]
Derivative of constant is zero
[katex]\boxed{\frac{dy}{dx}=0}[/katex]