Find the derivative of <msqrt> 5 + 4 cos &#x2061;<!-- ⁡ <mro

Yaritza Oneill

Yaritza Oneill

Answered question

2022-05-13

Find the derivative of 5 + 4 cos ( π x 5 ) .

Answer & Explanation

Makhi Lyons

Makhi Lyons

Beginner2022-05-14Added 15 answers

Use axn=axn to rewrite 5+4cos(πx5) as (5+4cos(πx5))12.

ddx[(5+4cos(πx5))12]

Differentiate using the chain rule, which states that ddx[f(g(x))] is f(g(x))g(x) where f(x)=x12 and g(x)=5+4cos(πx5).

12(5+4cos(πx5))12-1ddx[5+4cos(πx5)]

To write -1 as a fraction with a common denominator, multiply by 22.

12(5+4cos(πx5))12-122ddx[5+4cos(πx5)]

Combine -1 and 22.

12(5+4cos(πx5))12+-122ddx[5+4cos(πx5)]

Combine the numerators over the common denominator.

12(5+4cos(πx5))1-122ddx[5+4cos(πx5)]

Simplify the numerator.

12(5+4cos(πx5))-12ddx[5+4cos(πx5)]

Combine fractions.

12(5+4cos(πx5))12ddx[5+4cos(πx5)]

By the Sum Rule, the derivative of 5+4cos(πx5) with respect to x is ddx[5]+ddx[4cos(πx5)].

12(5+4cos(πx5))12(ddx[5]+ddx[4cos(πx5)])

Since 5 is constant with respect to x, the derivative of 5 with respect to x is 0.

12(5+4cos(πx5))12(0+ddx[4cos(πx5)])

Add 0 and ddx[4cos(πx5)].

12(5+4cos(πx5))12ddx[4cos(πx5)]

Since 4 is constant with respect to x, the derivative of 4cos(πx5) with respect to x is 4ddx[cos(πx5)].

12(5+4cos(πx5))12(4ddx[cos(πx5)])

Simplify terms.

222(5+4cos(πx5))12ddx[cos(πx5)]

Cancel the common factors.

2(5+4cos(πx5))12ddx[cos(πx5)]

Differentiate using the chain rule, which states that ddx[f(g(x))] is f(g(x))g(x) where f(x)=cos(x) and g(x)=πx5.

2(5+4cos(πx5))12(-sin(πx5)ddx[πx5])

Differentiate.

-2πsin(πx5)5(5+4cos(πx5))12


 

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