How do you differentiate #f(x)=(x^35x)^4#?
Use the chain rule,
Substituting this into the chain rule:
Reverse the u substitution:
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To differentiate ( f(x) = (x^3  5x)^4 ), you'll use the chain rule. The chain rule is a formula for computing the derivative of the composition of two or more functions. In this case, you have an outer function, ( g(u) = u^4 ), where ( u = x^3  5x ) is the inner function.
The chain rule states that if you have a composite function ( f(x) = g(h(x)) ), then the derivative ( f'(x) = g'(h(x)) \cdot h'(x) ).
Here, ( g(u) = u^4 ) and ( h(x) = x^3  5x ), so:

First, find the derivative of the outer function ( g(u) = u^4 ) with respect to ( u ), which is ( g'(u) = 4u^3 ).

Then, find the derivative of the inner function ( h(x) = x^3  5x ) with respect to ( x ), which is ( h'(x) = 3x^2  5 ).

Finally, apply the chain rule by multiplying these derivatives together, replacing ( u ) back with ( x^3  5x ):
[ f'(x) = 4(x^3  5x)^3 \cdot (3x^2  5) ]
So, the derivative ( f'(x) ) is ( 4(x^3  5x)^3(3x^2  5) ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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