# What is the derivative of #f(x)=ln[x^9(x+3)^6 (x^2+7)^5]#?

Use the properties of logarithms to rewrite:

So,

Simplify algebraically as desired.

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The derivative of ( f(x) = \ln[x^9(x+3)^6 (x^2+7)^5] ) is:

[ f'(x) = \frac{d}{dx}\left[\ln[x^9(x+3)^6 (x^2+7)^5]\right] ]

[ f'(x) = \frac{1}{x^9(x+3)^6 (x^2+7)^5} \times \frac{d}{dx}[x^9(x+3)^6 (x^2+7)^5] ]

[ f'(x) = \frac{1}{x^9(x+3)^6 (x^2+7)^5} \times \left[9x^8(x+3)^6(x^2+7)^5 + 6x^9(x+3)^5(x^2+7)^5 + 10x(x^2+7)^4(x+3)^6(x^2+7)^5 \right] ]

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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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