# How do you differentiate #sinx+xy+y^5=pi#?

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Differentiate each term:

The second term requires the use of the product rule:

The third term requires the use of the chain rule:

For the last term, we invoke the fact that the derivative of a constant is 0:

Divide both sides by the leading coefficient:

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To differentiate the equation ( \sin(x) + xy + y^5 = \pi ) with respect to ( x ), we use implicit differentiation. Taking the derivative of each term individually, we get:

( \frac{d}{dx}(\sin(x)) + \frac{d}{dx}(xy) + \frac{d}{dx}(y^5) = \frac{d}{dx}(\pi) )

Using the chain rule and product rule where necessary:

( \cos(x) + x\frac{dy}{dx} + y + 5y^4\frac{dy}{dx} = 0 )

Rearranging terms:

( (x + 5y^4)\frac{dy}{dx} = -\cos(x) - y )

Finally, solving for ( \frac{dy}{dx} ):

( \frac{dy}{dx} = \frac{-\cos(x) - y}{x + 5y^4} )

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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.

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.

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.

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