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hala718 eNotes educator| Certified Educator

f(x) = cos(x^2 + 3)

Let f(u) = cos(u)  such that:

u= x^2 + 2  ==> u' = 2x

==> f'(u) = -sin(u)

==> f'(x) = u'(x) * f'(u)

               = (2x)*(-sinu)

               = -2x*sinu

                 = -2x*sin(x^2+2)

==> f'(x) = -2x*sin(x^2 + 2)

neela | Student

To differentiate cos(x^2+3).

We know that d/dx u(v(x)) = d/dv(u) *dv/dx.

d/dx{ cos(x^2+3)} = d/dv(cosv) * dv/dx , where v = x^2+3

=-sinv * d/dx(x^2+2)

=-sin (x^2+2) * (2x)

d/dxcos(x^2+2) = -2x*cos(x^2+2)

william1941 | Student

To find the result for cos ( x^2 +3),

first equate y= x^2 +3 = y.

Now find the derivative of cos y and multiply that with the derivative of x^2+ 6.

We know that the derivative of cos ( y) is -sin y.

We also know that the he derivative of x^2+ 3 is 2x.

The derivative of cos ( x^2 +3) is [ - sin y * ( 2x) ]

= - sin ( x^2 +3)* 2x

= - 2x *sin ( x^2 +3)

The required answer is -2x sin (x^2+3)

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