What is the derivative of the square root of cos x?

Answer 1

To find the derivatve of the square root of cos x:

Use the chain rule; this means multiply the inner derivative by the outer derivative.

You can write the question f(x) = (cos x)1/2

This general break-down explains how to find d/dx f(x)

note: d/dx basically symbolizes "the derivative of"

In general terms:

f(x) = x1/2

g(x) = cos x

f(g(x)) = (cos x)1/2

outer derivative: d/dx f(z) = (1/2)*x-1/2 = 1/(4cos x)1/2

inner derivative: d/dx g(x) = -sin(x)

final answer: d/dx f(g(x)) = -sin(x)/(4*cos x)1/2

note: d/dx means "the derivative of"; so d/dx x = 1

Further explained:

Set up the equation to a more general form: (cos x)1/2

To make the inner derivative, look at cos(x)

To make the outer derivative, look at x1/2

note: x ~ cos x; so we treat (cos x) simply as x, to create the outer derivative

You probably know the necessary derivates:

1. derivative of cos x = -sin x

2. derivative of a1/2 = (1/2)*a-1/2 = 1/(4a)1/2

Multiplying the two we get the answer:

-sin(x)/(4cos x)1/2