How do you find the derivative of y= cos^3 w + cos(w^3)?

1 Answer
Mar 5, 2017

dy/dx = -3sinwcos^2w - 3w^2sin(w^3)

Explanation:

d/dx[a+b+c] = d/dx[a] + d/dx[b] + d/dx[c]

so

dy/dx = d/dx[cos^3w] + d/dx[cos(w^3)]

Now we've split it up, we can tackle each term separately.

The product rule states that

d/dx ab = bd/dx[a] + ad/dx[b]

so

d/dxcos^3w = coswd/dx[cos^2w] + cos^2wd/dxcosw

d/dxcos^2w = coswd/dxcosx + coswd/dxcosw

= -2sinwcosw

therefore,

d/dxcos^3w = cosw*-2sinwcosw + cos^2w*-sinw

= -3sinwcos^2w

Now we can begin to look at the second term, for which we need the chain rule:

d/dx f(g(x)) = g'(x)f'(g(x))

so

d/dx cos(w^3) = 3w^2 * -sin(w^3) = -3w^2sin(w^3)

Now we can put the whole thing back together,

dy/dx = -3sinwcos^2w-3w^2sin(w^3)