How do you implicitly derive #e^cos y=x^3 arctan y#?
then taking differential
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To implicitly derive ( e^{\cos y} = x^3 \arctan y ), you'll need to use the chain rule and product rule.
Starting with ( e^{\cos y} = x^3 \arctan y ):

Differentiate both sides with respect to ( x ): [ \frac{d}{dx} \left( e^{\cos y} \right) = \frac{d}{dx} \left( x^3 \arctan y \right) ]

Apply the chain rule on the left side: [ \frac{d}{dx} \left( e^{\cos y} \right) = e^{\cos y} \frac{d}{dx} (\cos y) ]

Apply the product rule on the right side: [ \frac{d}{dx} \left( x^3 \arctan y \right) = 3x^2 \arctan y + x^3 \frac{d}{dx} (\arctan y) ]

Compute ( \frac{d}{dx} (\cos y) ): [ \frac{d}{dx} (\cos y) = \sin y \frac{dy}{dx} ]

Compute ( \frac{d}{dx} (\arctan y) ): [ \frac{d}{dx} (\arctan y) = \frac{1}{1 + y^2} \frac{dy}{dx} ]

Substitute these derivatives back into the equation and solve for ( \frac{dy}{dx} ).
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When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
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