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Genrish500 [490]
3 years ago
15

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

Mathematics
1 answer:
Licemer1 [7]3 years ago
6 0

Answer:

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

Step-by-step explanation:

<u><em>Step(i):-</em></u>

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}

<u><em>Step(ii):-</em></u>

    Differentiating equation (i) with respective to 'x'

<em>First apply formula  </em>\frac{d}{dx} (x^{n} ) = n x ^{n-1}<em></em>

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

<em>Now we will apply formula </em>

     \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} )

<em>Again apply formula  </em>\frac{d}{dx} (x^{n} ) = n x ^{n-1}<em></em>

    \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}  )

<u><em>Final answer:-</em></u>

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

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