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If f[x] represents a function, its derivative is represented by f'[x]. Higher order derivatives are represented by f''[x], f'''[x], and so on.

f[x_]= x5 + x4 + x3 + x2 + x + 1; f'[x] 1 + 2 x + 3 x2 + 4 x3 + 5 x4 f''[x] 2 + 6 x + 12 x 2 + 20 x 3 f'''[x] 6 + 24 x + 60 x 2

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If a more traditional formatting of the derivatives is desired, the command TraditionalForm can be used.

Solving,

f[x_]= x5 + x4 + x3 + x2 + x + 1; f'[x] // TraditionalForm 5 x4 + 4 x3 + 3 x2 + 2 x + 1 f''[x] // TraditionalForm 20 x3 + 12 x2 + 6 x + 2 f'''[x] // TraditionalForm 60 x2 + 24 x + 6

The prime notation can also be used for built-in functions, as illustrated in the next example. If the argument is omitted, Mathematica returns a pure function representing the required derivative. (Pure functions are discussed in the appendix.)

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We have, of course, assumed that 1 8uv 6 0. Method 2: The given equations are F u2 v 3x y 0, G u 2v2 x 2y 0. Then by Problem 6.31,

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Sqrt' 1 & 2 #1 Sqrt'[x] 1 2 x Sqrt'' 1 & 4 #13/2

' Sqrt''[x]

1 4 x3/2

CHAP. 6]

D[f[x], x] returns the derivative of f with respect to x. D[f[x], {x, n}] returns the nth derivative of f with respect to x.

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D[x5 + x4 + x3 + x2 + x + 1, x] 1 + 2x + 3x 2 + 4x 3 + 5x 4 D[x5 + x4 + x3 + x2 + x + 1, {x, 2}] 2 + 6x + 12x 2 + 20x 3 D[x5 + x4 + x3 + x2 + x + 1, {x, 3}] 6 + 24x + 60x 2

, which can be found on the Basic Math Input palette, is equivalent to D. x will return the derivative with respect to x. The nth derivative is represented by {x, n}.

PARTIAL DERIVATIVES Fx @ F; G Gx @u @ x; v Fu @ F; G @x Gu @ u; v 3 1 Fv 1 4v 1 12v Gv 2u 1 1 8uv Fv 1 4v Gv

x(x5 + x4 + x3 + x2 + x + 1) 1 + 2x + 3x 2 + 4x 3 + 5x 4 {x, 2}(x5 + x4 + x3 + x2 + x + 1) 2 + 6x + 12x 2 + 20x 3 {x, 3}(x5 + x4 + x3 + x2 + x + 1) 6 + 24x + 60x 2

Derivative[n] is a functional operator that acts on a function to produce a new function, namely, its nth derivative. Derivative[n][f] gives the nth derivative of f as a pure function and Derivative[n][f][x]evaluates the nth derivative of f at x.

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It is useful to remember that f' is converted to Derivative[1]. Thus, f'[x] becomes Derivative[1][x]. Higher order derivatives f'', f''', etc. are handled in a similar manner.

6.33. If F u; v; w; x; y 0, G u; v; w; x; y 0, H u; v; w; x; y 0, nd @v ; b @x ; c @w : a @y x @v w @u y

f[x_] = x5 + x4 + x3 + x2 + x + 1; Derivative[1][f] 1 + 2#1 + 3#1 2 + 4#1 3 + 5#1 4 & Derivative[1][f][x] 1 + 2x + 3x 2 + 4x 3 + 5x 4

Mathematica returns a pure function representing the derivative of f. Pure functions are discussed in the appendix. #1 is replaced by x.

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