# The 50th derivative of y = cos 2 x . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 3.4, Problem 67E
To determine

## To find: The 50th derivative of y=cos2x.

Expert Solution

The 50th derivative of y=cos2x is d50ydx50=250cos2x_.

### Explanation of Solution

Given:

The function is y=cos2x.

Calculation:

The first derivative of y=cos2x is computed as follows,

y=ddx(y)=ddx(cos2x)=sin2x(2)=2sin2x

Thus, the first derivative of y=cos2x is y=2sin2x.

The second derivative of y is computed as follows,

y=ddx(y)=ddx(2sin2x)=2ddx(sin2x)

Simplify the terms,

y=2(dd(2x)(sin2x)d(2x)dx)=2(cos2x2)=22(cos2x)

Thus, the second derivative of y=cos2x is y=22(cos2x).

The third derivative of y is computed as follows,

y=ddx(y)=ddx(22(cos2x))=22ddx((cos2x))=22(sin2x2)

=23(sin2x)

Thus, the third derivative of y=cos2x is y=23(sin2x).

Proceed in the similar way, the 50th derivative y is computed as follows,

Since 50 = 12(4) + 2, the remainder is 2 and it corresponds with y.

d50ydx50=250cos2x

Therefore, the 50th derivative of y=cos2x is d50ydx50=250cos2x_.

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