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How do you determine dy/dx given #e^cosy=x^3siny#?

Answer 1

Ezekiel Alexander

The answer #[dy/dx]=[siny*3x^2]/[e^cosy*siny+x^3*cosy]#

#e^cosy=x^3*siny#

#-e^cosy*siny*(dy/dx)=x^3*cosy*(dy/dx)+siny*3x^2#

#e^cosy*siny*(dy/dx)+x^3*cosy*(dy/dx)=siny*3x^2#

#[dy/dx][e^cosy*siny+x^3*cosy]=siny*3x^2#

#[dy/dx]=[siny*3x^2]/[e^cosy*siny+x^3*cosy]#

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Answer 2

Ezekiel Alexander

To determine dy/dx given e^cosy = x^3siny, use implicit differentiation. Then, solve for dy/dx. The derivative of e^cosy with respect to y is -e^cosysin(y). The derivative of x^3siny with respect to y is x^3cos(y). Apply the chain rule when differentiating terms involving x. After finding dy/dx, isolate it to get the final answer.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.