Question

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

Answer 1

Answer:

$\frac{d y}{d x} = - (2 x + 3 x^{2} )sin2(x^{2} +x^{3})$

Step-by-step explanation:

Step(i):-

Given y = cos² (x² + x³)  ....(i)

By using differentiation formulas

a)    $\frac{d}{dx} (cosx) = -sinx$

b)    $\frac{d}{dx} (x^{n} ) = n x ^{n-1}$

Step(ii):-

    Differentiating equation (i) with respective to 'x'

First apply formula  $\frac{d}{dx} (x^{n} ) = n x ^{n-1}$

      $\frac{d y}{d x} = 2 cos (x^{2} +x^{3} )^{2-1} \frac{d}{d x} (cos(x^{2} +x^{3})$      

Now we will apply formula

     $\frac{d}{dx} (cosx) = -sinx$

     $\frac{d y}{d x} = 2 cos (x^{2} +x^{3} ) (-sin(x^{2} +x^{3})\frac{d}{dx} (x^{2} +x^{3} )$

Again apply formula  $\frac{d}{dx} (x^{n} ) = n x ^{n-1}$

    $\frac{d y}{d x} = 2 cos (x^{2} +x^{3} ) (-sin(x^{2} +x^{3}) (2 x + 3 x^{2} )$

  $\frac{d y}{d x} = -2 sin (x^{2} +x^{3} ) cos(x^{2} +x^{3}) (2 x + 3 x^{2} )$

we know that trigonometric formulas

Sin 2θ  = 2 sinθ cosθ

$\frac{d y}{d x} = -sin2(x^{2} +x^{3}) (2 x + 3 x^{2} )$

Final answer:-

$\frac{d y}{d x} = - (2 x + 3 x^{2} )sin2(x^{2} +x^{3})$