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Lana71 [14]
2 years ago
15

Please help me answer this question

Mathematics
1 answer:
stealth61 [152]2 years ago
8 0

The value of dy/dx for the functions are

(i) \frac{dy}{dx} = 4x^{2}sin2x.cos2x+ 2x. sin^{2}2x

(ii) \frac{dy}{dx} =\frac{- y(1+3x^{2})}{2x(1+x^{2}) }

<h3>Differentiation</h3>

From the question, we are to determine dy/dx for the given functions

(i) x^{2} sin^{2}2x

Let u = x^{2}

and

v = sin^{2} 2x

Also,

Let w=  sin2x

∴ v = w^{2}

First, we will determine \frac{dv}{dx}

Using the Chain rule
\frac{dv}{dx} = \frac{dv}{dw}.\frac{dw}{dx}

v = w^{2}

∴ \frac{dv}{dw} =2w

Also,

w=  sin2x

∴ \frac{dw}{dx} =2cos2x

Thus,

\frac{dv}{dx} = 2w \times 2cos2x

\frac{dv}{dx} = 2sin2x \times 2cos2x

\frac{dv}{dx} = 4sin2x . cos2x

Now, using the product rule

\frac{dy}{dx} = u\frac{dv}{dx} +  v\frac{du}{dx}

From above

u = x^{2}

∴ \frac{du}{dx}=2x

Now,

\frac{dy}{dx} = x^{2} (4sin2x.cos2x)+  sin^{2}2x (2x)

\frac{dy}{dx} = 4x^{2}sin2x.cos2x+ 2x. sin^{2}2x

(ii) xy^{2}+y^{2}x^{3} +2=0

Then,

x.2y\frac{dy}{dx}+ y^{2}(1)+y^{2}.3x^{2} + x^{3}.2y\frac{dy}{dx} +0=0

2xy\frac{dy}{dx}+ y^{2}+3x^{2}y^{2} + 2x^{3}y\frac{dy}{dx} =0

2xy\frac{dy}{dx}+2x^{3}y\frac{dy}{dx} =-  y^{2}-3x^{2}y^{2}

\frac{dy}{dx} (2xy+2x^{3}y)=-  y^{2}(1+3x^{2})

\frac{dy}{dx} =\frac{- y^{2}(1+3x^{2})}{2xy+2x^{3}y}

\frac{dy}{dx} =\frac{- y^{2}(1+3x^{2})}{2xy(1+x^{2}) }

\frac{dy}{dx} =\frac{- y(1+3x^{2})}{2x(1+x^{2}) }

Hence, the value of dy/dx for the functions are

(i) \frac{dy}{dx} = 4x^{2}sin2x.cos2x+ 2x. sin^{2}2x

(ii) \frac{dy}{dx} =\frac{- y(1+3x^{2})}{2x(1+x^{2}) }

Learn more on Differentiation here: brainly.com/question/24024883

#SPJ1

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