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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
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