# How do you differentiate #g(x) = sqrt(1-x)cosx# using the product rule?

According to the product rule,

In this case, we have

First, ascertain each of these functions' respective derivatives.

Re-enter these into the expression of the product rule.

This is easier to understand:

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To differentiate ( g(x) = \sqrt{1-x} \cos x ) using the product rule, you would follow these steps:

- Identify the two functions: ( f(x) = \sqrt{1-x} ) and ( h(x) = \cos x ).
- Apply the product rule formula: ( (f \cdot h)' = f' \cdot h + f \cdot h' ).
- Find the derivatives of ( f(x) ) and ( h(x) ).
- ( f'(x) = -\frac{1}{2\sqrt{1-x}} ) (using the chain rule)
- ( h'(x) = -\sin x ) (derivative of ( \cos x ))

- Substitute the derivatives into the product rule formula.
- ( g'(x) = -\frac{1}{2\sqrt{1-x}} \cdot \cos x + \sqrt{1-x} \cdot (-\sin x) )

- Simplify the expression if necessary.

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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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