Question

Differentiate... How to solve this type of problem? y = cos^2(x^2 + x^3)

Answer 1

`(d y)/(d x) = - (2 x + 3 x^(2) )sin2(x^(2) +x^(3))`

Explanation:

Step(i):-

Given y = cos² (x² + x³) ....(i)

By using differentiation formulas

a)
`(d)/(dx) (cosx) = -sinx`

b)
`(d)/(dx) (x^(n) ) = n x ^(n-1)`

Step(ii):-

Differentiating equation (i) with respective to 'x'

First apply formula
`(d)/(dx) (x^(n) ) = n x ^(n-1)`

`(d y)/(d x) = 2 cos (x^(2) +x^(3) )^(2-1) (d)/(d x) (cos(x^(2) +x^(3))`

Now we will apply formula

`(d)/(dx) (cosx) = -sinx`

`(d y)/(d x) = 2 cos (x^(2) +x^(3) ) (-sin(x^(2) +x^(3))(d)/(dx) (x^(2) +x^(3) )`

Again apply formula
`(d)/(dx) (x^(n) ) = n x ^(n-1)`

`(d y)/(d x) = 2 cos (x^(2) +x^(3) ) (-sin(x^(2) +x^(3)) (2 x + 3 x^(2) )`

`(d y)/(d x) = -2 sin (x^(2) +x^(3) ) cos(x^(2) +x^(3)) (2 x + 3 x^(2) )`

we know that trigonometric formulas

Sin 2θ = 2 sinθ cosθ

`(d y)/(d x) = -sin2(x^(2) +x^(3)) (2 x + 3 x^(2) )`

Final answer:-

`(d y)/(d x) = - (2 x + 3 x^(2) )sin2(x^(2) +x^(3))`