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How do you differentiate #f(x)=1/(2x+5)^5 # using the chain rule?

Answer 1

Paisley Johnson

#-10/(2x+5)^6#

The equation can be rewritten as #(2x+5)^-5#. Now, use chain rule to differentiate:

#d/dx(f(g(x)))=f'(g(x))*g'(x)#

#-5(2x+5)^-6*2#

Simplify:

#-10 * (2x+5)^-6#

#-10/(2x+5)^6#

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Answer 2

Ezra Adams

To differentiate ( f(x) = \frac{1}{(2x+5)^5} ) using the chain rule, follow these steps:

Identify the inner function: ( u = 2x + 5 ).
Find the derivative of the inner function with respect to ( x ): ( \frac{du}{dx} = 2 ).
Apply the chain rule: ( \frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} ).
Differentiate the outer function with respect to ( u ): ( \frac{df}{du} = -5(2x+5)^{-6} ).
Multiply the result from step 4 by the result from step 2 to get the final answer: [ \frac{df}{dx} = -5(2x+5)^{-6} \cdot 2 ] [ = -10(2x+5)^{-6} ]

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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.