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Find the value of dy/dx at θ=pi/4 if x=ae^θ (sinθ-cosθ) and y=ae^θ(sinθ+cosθ) - Mathematics and Statistics

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Find the value of `dy/dx " at " theta =pi/4 if x=ae^theta (sintheta-costheta) and y=ae^theta(sintheta+cos theta)`

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Solution

`y=ae^theta(sintheta+cos theta)`

`x=ae^theta (sintheta-costheta)`

Differentiating y with respect to θ on both the sides, we get:

`dy/(d theta)=ae^theta(costheta-sintheta)+ae^theta(sintheta+costheta)dy/(d theta)`

`=2ae^theta cos theta`

Differentiating x with respect to θ on both the sides, we get:

`dx/(d theta)=ae^theta(costheta+sintheta)+ae^theta(sintheta-costheta)dx/(d theta)`

`=2ae^theta sin theta`

Now

`dy/dx=(dy/(d theta))/(dx/(d theta))=(2ae^theta cos theta)/(2ae^theta sin theta)=cot theta`

`(dy/dx)_(theta=pi/4)=cot(pi/4)=1`

Concept: Derivatives of Functions in Parametric Forms
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