# What is the derivative of #y = ln ((x^2+1)^5 / sqrt(1-x)) #?

As an alternative to Stefan's solution (which is a fine solution), we could use the properties of logarithms to rewrite

(Showing that his is the same as Stefan's answer is left as an algebra exercise.)

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You're going to have to use thw whole arsenal for this one - chain rule, power rule, quotient rule.

Since you can write your function as

you can use the chain rule to differentiate it like this

You will get

This can be further broken down into

Plug this back into your target derivative to get

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The derivative of (y = \ln\left(\frac{(x^2+1)^5}{\sqrt{1-x}}\right)) with respect to (x) is:

[y' = \frac{d}{dx} \ln\left(\frac{(x^2+1)^5}{\sqrt{1-x}}\right) = \frac{d}{dx} \left(5 \ln(x^2 + 1) - \frac{1}{2} \ln(1-x)\right)]

[y' = 5 \cdot \frac{d}{dx} \ln(x^2 + 1) - \frac{1}{2} \cdot \frac{d}{dx} \ln(1-x)]

Using the chain rule and the derivative of (\ln(u)), where (u) is a function of (x):

[y' = 5 \cdot \frac{1}{x^2 + 1} \cdot 2x + \frac{1}{2(1-x)}]

[y' = \frac{10x}{x^2 + 1} + \frac{1}{2(1-x)}]

Simplify this expression to get the final derivative.

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