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