# Differentiate? #F(y) = (1/y^2 - 5/y^4) (y +7y^3) #

Apply the rule of the product:

Perform the initial derivative:

Return to the product rule by substituting:

Compute the second derivative:

Return to the product rule by substituting:

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To differentiate the function ( F(y) = \left(\frac{1}{y^2} - \frac{5}{y^4}\right) \cdot (y + 7y^3) ), you can use the product rule and the power rule.

First, apply the product rule: [ F'(y) = \left(\frac{1}{y^2} - \frac{5}{y^4}\right) \cdot (1 + 21y^2) + \left(\frac{1}{y^2} - \frac{5}{y^4}\right) \cdot (y + 7y^3)' ]

Now, differentiate each term using the power rule: [ F'(y) = \left(\frac{1}{y^2} - \frac{5}{y^4}\right) \cdot (1 + 21y^2) + \left(\frac{1}{y^2} - \frac{5}{y^4}\right) \cdot (1 + 21y^2) ]

Finally, simplify the expression: [ F'(y) = \left(\frac{1}{y^2} - \frac{5}{y^4}\right) \cdot (1 + 21y^2) + \left(\frac{1}{y^2} - \frac{5}{y^4}\right) \cdot (1 + 21y^2) ]

[ F'(y) = \frac{1}{y^2} + 21 - \frac{5}{y^4} - \frac{105}{y^2} + \frac{1}{y^2} + 21 - \frac{5}{y^4} - \frac{105}{y^2} ]

[ F'(y) = \frac{2}{y^2} + 42 - \frac{10}{y^4} - \frac{210}{y^2} ]

So, the derivative of ( F(y) ) with respect to ( y ) is ( F'(y) = \frac{2}{y^2} + 42 - \frac{10}{y^4} - \frac{210}{y^2} ).

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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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