# War & Peace

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Derivative
The derivative of the function f with respect to the variable x is the function fı whose value at x is fıx= limh→0fx+h-f(x)h provided the limit exist.
Derivative
The derivative of the function f with respect to the variable x is the function fı whose value at x is fıx= limh→0fx+h-f(x)h provided the limit exist.
Differentiation
f(x) | ddx | xn | nxn-1 | f∙g | fgı+gfı | fg | gfı- fgı g2 | f(gx) | fı(gx)gı(x) | sinx | cosx | cosx | -sinx | tanx | sec2x | cotx | -csc2x | secx | secxtanx | cscx | -cscxcotx | ex | ex | ax | axlna | lnx | 1x | Arcsin x | 11-x2 | Arctan x | 11+x2 |
The Sum and Difference Rule
If u and v are differentiable functions ofx, then their sum and difference are differentiable at every point where u and v are differentiable. At such points, ddx(u±v)= dudx ±dvdx
The Sum and Difference Rule
If u and v are differentiable functions ofx, then their sum and difference are differentiable at every point where u and v are differentiable. At such points, ddx(u±v)= dudx ±dvdx
The Constant Multiple Rule
If u is a differentiable function of x and c is a constant, then ddx(cu)= cdudx
The Constant Multiple Rule
If u is a differentiable function of x and c is a constant, then ddx(cu)= cdudx
Power Rule for Positive Integer Powers of x
If n is a positive integer, then ddx(xn)= nxn-1
Power Rule for Positive Integer Powers of x
If n is a positive integer, then ddx(xn)= nxn-1
Derivative of Constant Function
If f is the function with the constant value of c, then dfdx= ddx c=0
Derivative of Constant Function
If f is the function with the constant value of c, then dfdx= ddx c=0
One-Sided Derivatives
A function y=f(x) is differentiable on a closed interval [a,b] if it has a derivative at every interior point of the interval, and if the limits limh→0+fa+h-f(a)h [the right-hand derivative

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