How do you differentiate #f(x)=(x^3-5x)^4#?

Answer 1

Use the chain rule, #(df)/dx = (df)/(du)(du)/(dx)#

Let #u = x^3 - 5x#, then, #f(u) = u^4#, #(df)/(du) = 4u^3#, and #(du)/(dx) = 3x² - 5#

Substituting this into the chain rule:

#(df)/dx = 4u³(3x² - 5)#

Reverse the u substitution:

#(df)/dx = 4(x^3 - 5x)³(3x² - 5)#
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Answer 2

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:

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

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

  3. 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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Answer from HIX Tutor

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.

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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