Find the derivative of the function. Simplify if possible. and...... y = 17 arctan(sqrt x) y = arcsin(4x + 2) y = arccos(e8x) h(t) = 5arccot(t) + 5arccot(1/t)

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neela | High School Teacher | (Level 3) Valedictorian

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To find the derivative of 

1)y = 17 arctan sqrtx

tan (y/17) =  sqrtx

Differentiating with respect to x, weget:

sec^2 (y/17) * (1/17) dy/dx = (1/2sqrtx)

dy/ dx = (17/2sqrtx){1+(tan(y/17))^2}

dy/dx = (17/2sqrtx) {1+x}

 

2)y = arc sin(4x+2)

Therefore siny = 4x+2

Differentiate both sides:

cosy*dy/dx = d/dx(4x+2)

cosy * dy/dx = 4

dy/dx = 4/cosy = 4/sin^2y = 4/(4x+2)^2

dy/dx = 4/(4x+2)^2.

 

3)y=arccos(e^8x).

Therefore cosy = e^(8x).

Differentiating both sodes, we get:

-siny*dy/dx= d/dx{e^(8x)}.

-siny*dy/dx = e^(8x)* d/dx(8x).

-siny*dy/dx = 8e^(8x).

dy/dx = 8e^(8x)* (1/-siny)

dy/dx = -8e^(8x)/ (1-(cosy)^2)

 dy/dx = -8e^(8x)/{1- e^(16x)}, as (cosy)^2 = (e^(8x))^2.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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1) To calculate the derivative of y = sqrt(3arctan x),we'll apply the chain rule:

y' = [sqrt(3arctan x)]'*(3arctan x)'

We'll see 3arctan x as an entity and we'll differentiate the sqrt:

(sqrt t)' = 1/2sqrt t

y' = [1/2sqrt(3arctan x)]*[3/(1+x^2)]

y' = 3/[2*(1+x^2)*sqrt(3arctan x)]

2) We'll calculate the derivative of  y = 17 arctan (sqrt x) using the chain rule also:

y' = [17 arctan (sqrt x)]'*(sqrt x)'

y' = [17/(1+(sqrtx)^2)]*(1/2sqrtx)

y' = 17/2(sqrtx)*(1+x)

3) We'll calculate the derivative of  y = arcsin(4x + 2)using the chain rule also:

y = [arcsin(4x + 2)]'*(4x+2)'

We'll see 4x + 2 as an entity:

y' = {1/sqrt[1-(4x+2)^2]}*(4)

We'll expand the square:

y' = 4/sqrt(1-16x^2 - 8x - 4)

We'll combine like terms:

y' = 4/sqrt(-3-16x^2 - 8x)

4) We'll calculate the derivative of  y = arccos(e^8x) using the chain rule also:

y = arccos(e^8x)

We'll put (e^8x) = t

(arccos t)' = 1/sqrt(1-t^2)

y' = [arccos(e^8x)]'*(e^8x)*(8x)'

y' = [1/sqrt(1-(e^8x)^2)]*(e^8x)*(8)

y' = 8(e^8x)/[sqrt(1-(e^8x)^2]

5) We'll calculate the derivative of  y = arctan [x+sqrt(x^2+1)] using the chain rule also:

y = arctan [x+sqrt(x^2+1)]

We'll put x+sqrt(x^2+1) = t

(arctan t)' = 1/(1+t^2)

y' = {arctan [x+sqrt(x^2+1)]}'*[x+sqrt(x^2+1)]'

y' = {1/{1+[x+sqrt(x^2+1)]^2}}*[1 + 2x/2sqrt(x^2+1)]

6) We'll calculate the derivative of  y =  5arccot(t) + 5arccot(1/t)using the chain rule also:

h(t) = 5arccot(t) + 5arccot(1/t)

h'(t) = -5/(1+t^2) - 5/[1+(1/t)^2]

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