   Chapter 11.3, Problem 56E 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

In Exercises 29-56, calculate d y d x . You need not expand your answers. [HINT: See Example 1 and 2.] y = 3 x − 1 ( x − 5 ) ( x − 4 ) ( x − 1 )

To determine

To calculate: The derivative of function y=3x1(x5)(x4)(x1).

Explanation

Given Information:

The function is y=3x1(x5)(x4)(x1).

Formula used:

Quotient rule:

If f(x) and g(x) are differentiable functions, then

ddx[f(x)g(x)]=f'(x)g(x)f(x)g'(x)[g(x)]2 where, g(x)0.

Product rule:

If f(x), g(x) and h(x) are differentiable functions, then

(fgh)'=f'gh+fg'h+fgh'

Power rule:

For a function f(x)=xn,

f'(x)=nxn1, where n is some constant.

Calculation:

Consider the function, y=3x1(x5)(x4)(x1)

Apply quotient rule of derivative,

dydx=[ddx(3x1)](x5)(x4)(x1)(3x1)[ddx[(x5)(x4)(x1)]][(x5)(x4)(x1)]2. …… (1)

Now, apply product rule to the product (x5)(x4)(x1),

ddx(x5)(x4)(x1)=[ddx(x5)](x4)(x1)+(x5)[ddx(x4)](x1)+(x5)(x4)[ddx(x1)]

Apply power rule and simplify further,

ddx(x5)(x4)(x1)=[1x115(0)](x4)(x1)+(x5)[1x114(0)](x1)+(x5)(x4)[1x111(0)]=(x0)(x4)(x1)+(x5)(x0)(x1)+(x5)(x4)x0=[x(x)+x(1)4(x)4(1)]+[x(x)+x(1)5(x)5(1)]+[x(x)+x(4)5(x)5(4)]=(x25x+4)+(x26x+5)+(x29x+20)

Combine like terms,

ddx(x5)(x

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