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EastWind [94]
3 years ago
5

Use logarithmic differentiation to find the derivative of the function. y = x2cos x Part 1 of 4 Using properties of logarithms,

we can rewrite ln y = ln(x2cos x) as
Mathematics
1 answer:
Arisa [49]3 years ago
7 0

ANSWER

{y}^{'}  = 2x \cos(x)  -   {x}^{2} \sin(x)

EXPLANATION

The given function is

y =  {x}^{2}  \cos(x)

We take natural log of both sides;

ln(y) =   ln({x}^{2}  \cos(x) )

Recall and use the product rule of logarithms.

ln(AB)  =  ln(A )  +  ln(B)

This implies that:

ln(y) =   ln({x}^{2}  ) +  ln( \cos(x) )

ln(y) =  2 ln({x} ) +  ln( \cos(x) )

We now differentiate implicitly to obtain;

\frac{ {y}^{'} }{y}  =  \frac{2}{x}   -  \frac{ \sin(x) }{ \cos(x) }

Multiply through by y,

{y}^{'} = y( \frac{2}{x}   - \frac{ \sin(x) }{ \cos(x) ) })

Substitute y=x²cosx to obtain;

{y}^{'} =  {x}^{2}  \cos(x) ( \frac{2}{x}   - \frac{ \sin(x) }{ \cos(x) ) } )

Expand:

{y}^{'}  = 2x \cos(x)  -   {x}^{2} \sin(x)

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