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How do you use the chain rule to differentiate #y=4/(sqrt(x-5)#?

Answer 1

Charles Arocha

#y=4/(sqrt(x-5))#

We can rewrite this as: #y=4(x-5)^(-1/2)#

Use the chain rule: #y'=4(-1/2)(x-5)^(-3/2)(1)#

#y'=-2(x-5)^(-3/2)#

#y'=(-2)/(x-5)^(3/2)#

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Answer 2

Ezekiel Alexander

To differentiate ( y = \frac{4}{\sqrt{x-5}} ) using the chain rule:

Identify the outer function and the inner function.
Apply the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

Let ( u = x - 5 ).

Rewrite the function in terms of ( u ): ( y = 4u^{-\frac{1}{2}} ).
Differentiate the function ( y ) with respect to ( u ): ( \frac{dy}{du} = -2u^{-\frac{3}{2}} ).
Now, differentiate ( u ) with respect to ( x ): ( \frac{du}{dx} = 1 ).
Apply the chain rule: ( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} ).
Substitute ( \frac{dy}{du} ) and ( \frac{du}{dx} ) into the equation to get the final result.

[ \frac{dy}{dx} = -2(x - 5)^{-\frac{3}{2}} ]

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Answer from HIX Tutor

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