Derivative

Author

Eric W. Weisstein
January 14, 2007

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/Derivative.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Derivative.html.

©2007 Wolfram Research, Inc. except for portions noted otherwise

Mathematica Syntax

Derivative[2][#^5&]

Derivative[2][#^5&][2]

Definition

First Derivative

Limit[((f[x+h]-f[x])/h),h->0,Analytic->True]

Limit[((f[x+h/2]-f[x-h/2])/h),h->0,Analytic->True]

Second Derivative

Limit[((f[x+2h]-2f[x+h]+f[x])/(h^2)),h->0,Analytic->True]

f[x_]:=Piecewise[{{x^2,x≥0},{-x^2,x<0}}]

D[f[x],{x,2}]

Limit[((f[x+2h]-2f[x+h]+f[x])/(h^2)),h->0]

Multiple Derivatives of Powers Times a Functions

(d=Table[D[x^k f[x],{x,k}],{k,0,5}])//ColumnForm

CoefficientList[#,x]&/@(d/.f[x]:>1/.Derivative[_][_][_]:>1)

l=Table[LaguerreL[n,x],{n,0,5}]

Abs[MapIndexed[CoefficientList[#1,x]Sequence@@(#2-1)!&,l]]

(((∂^2)y(x))/(∂(x^2)))==y(x)

DSolve[y'[x]==y[x],y[x],x]

DSolve[y''[x]==y[x],y[x],x]

DSolve[y'''[x]==y[x],y[x],x]

DSolve[y''''[x]==y[x],y[x],x]

DSolve[y'''''[x]==y[x],y[x],x]

DSolve[D[y[x],{x,6}]==y[x],y[x],x]

DSolve[D[y[x],{x,7}]==y[x],y[x],x]

DSolve[D[y[x],{x,8}]==y[x],y[x],x]

FoxTrot

FullSimplify[((u^(n+1))/(((n+1))^2))((n+1)Log[u]-1)]

D[((u^(n+1))/(((n+1))^2))((n+1)Log[u]-1),u]//FullSimplify

Integrate[u^n Log[u],u]//TraditionalForm