Better Students Ask More Questions.
DerivativeDetermine the derivative of - cosx - cos^3x/3
2 Answers | add yours
We'll note the given expression with the function f(x).
f(x) = - cosx - cos^3x/3
We'll differentiate f(x) with respect to x:
f'(x) = [- cosx - (cosx)^3/3]'
f'(x) = (- cosx)' + [- (cosx)^3/3]'
f'(x) = -(-sin x) - 3(cosx)^2*(-sin x)/3
We'll simplify and we'll get:
f'(x) = sin x + (cosx)^2*(sin x)
We'll substitute (cosx)^2 = 1 - (sin x)^2:
f'(x) = sin x + [1 - (sin x)^2]^2*(sin x)
We'll remove the brackets:
f'(x) = sin x + sin x - (sin x)^3
We'll combine like terms:
f'(x) = 2sin x - (sin x)^3
Posted by giorgiana1976 on June 4, 2011 at 4:16 AM (Answer #2)
You need to find the derivative of the function using the derivative of composed function for the member `(cos^3 x)/3` , such that:
`(d(f(x)))/(dx) = - (d(cos x))/(dx) - (d(cos^3 x))/3)'`
`(d(f(x)))/(dx)= sin x + 3(cos^2 x)*(sin x)/3`
Reducing duplicate factors yields:
`(d(f(x)))/(dx) = sin x + cos^2 x*sin x`
Factpring out `sin x` yields:
`(d(f(x)))/(dx) = sin x*(1 + cos^2 x)`
Hence, evaluating the derivative of the given trigonometric function yields `(d(f(x)))/(dx) = sin x*(1 + cos^2 x).`
Posted by sciencesolve on April 30, 2013 at 6:17 PM (Answer #3)
Join to answer this question
Join a community of thousands of dedicated teachers and students.