Verified answer

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f(x) = cos( sin( x² ))

f'(x) = -sin( sin( x² )) • cos( x² ) • 2x            ← by the Chain Rule

       = -2x sin( sin( x² )) cos( x² )

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EDIT

The "chain" is:   cos( sin( x² )) ... sin( x² ) ... x²

So, in words, the derivative of cos( sin( x² )) with respect to x is:

i. The derivative of cos( sin( x² )) with respect to sin( x² ) which is -sin( sin( x² ))

           (Just as the derivative of cosθ with respect to θ is  -sinθ )

TIMES

ii. The derivative of sin( x² ) with respect to x² which is cos(x²)

       (Just as the derivative of sinθ with respect to θ is cosθ)

TIMES

iii. The derivative of x² with respect to x ... which is 2x

That makes:

d cos( sin( x² ))

———————  =  -sin( sin( x² )) • cos( x² ) • 2x

         dx

Hope that helps to eliminate the confusion.

Have a good one!!

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Yes you need the chain rule.

#1 - Differentiate cos

#2 - Differentiate sin

#3 - Differentiate x^2

=-sin(sin(x^2) cos(x^2) 2x

diff(cos(sin(x^2)), x) = - 2*sin(sin(x^2))*cos(x^2)*x