# How do you differentiate #f(x)=x^3*sqrt(x-2)-sinxcosx# using the product rule?

Starting with the first of the two products,

the product rule states

Detour for chain rule applied to the second function:

Denoting

under the chain rule,

so that

Also

so

or, reverting to square root notation,

Returning to the application of the product rule

On to the second of the two products.

as already noted, under the product rule,

so

that is

So, the overall derivative (under the sum rule) is

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To differentiate ( f(x) = x^3\sqrt{x-2} - \sin(x)\cos(x) ) using the product rule, first, identify the functions being multiplied together. Then, apply the product rule which states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

Let ( u = x^3 ) and ( v = \sqrt{x-2} ), and ( w = \sin(x) ) and ( z = \cos(x) ).

Then, differentiate each function with respect to ( x ):

( u' = 3x^2 ) (derivative of ( x^3 ))

( v' = \frac{1}{2\sqrt{x-2}} ) (derivative of ( \sqrt{x-2} ))

( w' = \cos(x) ) (derivative of ( \sin(x) ))

( z' = -\sin(x) ) (derivative of ( \cos(x) ))

Now, apply the product rule:

( f'(x) = u'v + uv' - w'z - wz' )

( f'(x) = (3x^2)(\sqrt{x-2}) + (x^3)\left(\frac{1}{2\sqrt{x-2}}\right) - (\sin(x))(-\sin(x)) - (\cos(x))(\cos(x)) )

Simplify the expression to obtain the derivative of ( f(x) ).

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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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