   Chapter 11.1, Problem 72E Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Solutions

Chapter
Section Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

Write out the proof that d d x ( x 5 ) = 5 x 4

To determine

To prove: The expression ddx(x5)=5x4.

Explanation

Formula used:

Derivative of function f(x) is computed by the use of the formula f'(x)=limh0f(a+h)f(a)h.

Binomial expansion of function is (x+a)n=k=0nnCkxka(nk).

Proof:

Consider the expression, ddx(x5)=5x4

Apply definition of the derivative to function x5.

ddx(x5)=limh0[(x+h)5][x5]h

Use binomial expansion to solve (x+h)5,

(x+h)5=k=055Ckxkh(5k)=5C0x0h(50)+5C1x1h(51)+5C2x2h(52)+5C3x3h(53)+5C4x4h(54)+5C5x5h(55)=1h5+5xh4+10x2h3+10x3h2+5x4h+1x5=h5+5xh4+10x2h3+10x3h2+5x4

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