How do you use the chain rule to differentiate #y=7/(2x+7)^2#?

Answer 1

#dy/dx=(-28)/(2x+7)^3#

The derivative of #x^-2# is #-2x^-3#, through the power rule.
When we have a function to the negative second, we take its derivative the same way, but then multiply that by the derivative of the inner function through the chain rule. That is, the derivative of #(f(x))^-2# is, through the chain rule, #-2(f(x))^-3*f'(x)#.

This said, we have:

#y=7(2x+7)^-2#

So its derivative is:

#dy/dx=7(-2(2x+7)^-3)*d/dx(2x+7)#
Don't forget to multiply by the derivative of #2x+7#, which is #2#.
#dy/dx=-28(2x+7)^-3#
#dy/dx=(-28)/(2x+7)^3#
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Answer 2

To differentiate ( y = \frac{7}{(2x + 7)^2} ) using the chain rule, follow these steps:

  1. Identify the outer function ( f(u) ) and the inner function ( u(x) ).

    • The outer function is ( f(u) = u^2 ).
    • The inner function is ( u(x) = 2x + 7 ).
  2. Compute the derivative of the outer function with respect to ( u ), denoted as ( f'(u) ), and the derivative of the inner function with respect to ( x ), denoted as ( u'(x) ).

    • ( f'(u) = 2u )
    • ( u'(x) = 2 )
  3. Substitute the inner function into the derivative of the outer function:

    • ( \frac{d}{dx} [f(u)] = f'(u) \cdot u'(x) = 2u \cdot 2 )
  4. Substitute the inner function ( u(x) = 2x + 7 ) back into the result:

    • ( \frac{d}{dx} [f(u)] = 2(2x + 7) )
  5. Simplify the expression:

    • ( \frac{d}{dx} [f(u)] = 4x + 14 )

Therefore, the derivative of ( y = \frac{7}{(2x + 7)^2} ) with respect to ( x ) is ( \frac{d}{dx} [y] = 4x + 14 ).

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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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