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How do you differentiate #f(x) = -15 / (4x + 5)^4#?

Answer 1

Luke Alvarez

#f'(x)=240/(4x+5)^5#

As #f(x)=-15/(4x+5)^4=-15*(4x+5)^(-4)#

We can do a chain differentiation #f'(x)=-15*((4x+5)^(-4))'*(4x)'#

#=-15*-4(4x+5)^(-5)*4#

#=240/(4x+5)^(5)#

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Answer 2

Charles Anctil

To differentiate the function ( f(x) = -\frac{15}{(4x + 5)^4} ), you can use the chain rule. The chain rule states that if you have a function within another function, you differentiate the outer function first, then multiply it by the derivative of the inner function.

Here's the process:

Differentiate the outer function: [ f'(x) = -15 \cdot \frac{d}{dx}\left((4x + 5)^{-4}\right) ]
Differentiate the inner function: [ \frac{d}{dx}\left((4x + 5)^{-4}\right) = -4(4x + 5)^{-5} \cdot \frac{d}{dx}(4x + 5) ]
Compute the derivative of ( 4x + 5 ): [ \frac{d}{dx}(4x + 5) = 4 ]
Substitute the derivative of the inner function into the expression: [ f'(x) = -15 \cdot (-4)(4x + 5)^{-5} \cdot 4 ]
Simplify: [ f'(x) = \frac{240}{(4x + 5)^{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.