# How do you differentiate #y= 1/x (sin^-5(x)) - x/3 (cos^3(x))#?

By saying

We can break it down into smaller derivatives and doing that such that

And for

We have, by the product rule and the chain rule

Adding it all up we have

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To differentiate the given function ( y = \frac{1}{x} \sin^{-5}(x) - \frac{x}{3} \cos^3(x) ):

- Differentiate each term separately using the product rule and chain rule where necessary.
- The derivative of (\frac{1}{x}\sin^{-5}(x)) can be calculated using the product rule and chain rule.
- The derivative of (\frac{x}{3}\cos^3(x)) can also be calculated using the product rule and chain rule.
- Combine the derivatives of each term to obtain the overall derivative of the function.

The derivative of the given function is:

[ y' = -\frac{\sin^4(x)}{x^2} - \frac{1}{x} \cdot 5 \sin^{-6}(x) \cos(x) - \frac{1}{3} \cos^2(x) + \frac{x}{3} \cdot 3 \cos^2(x) \sin(x) ]

Simplify the expression further if needed.

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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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