# How do you differentiate #(3x-4/x)^2 (1-x+7x^2)^4#?

Chain rule and product rule time combined!

First, we will rename:

Using the product rule, as follows,

Now, we will lead with the colored derivatives first and then return to the whole equation above.

You may want to expand it and eliminate the factoring (parenthesis), but that is not really necessary as the work and steps to achieve so would not necessarily mean simplification. The line above can be your answer.

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To differentiate the expression ( (3x - \frac{4}{x})^2 (1 - x + 7x^2)^4 ), you can use the product rule. The product rule states that if you have two functions, ( u ) and ( v ), then the derivative of their product is given by ( u'v + uv' ). Here's how you apply it to this expression:

Let ( u = (3x - \frac{4}{x})^2 ) and ( v = (1 - x + 7x^2)^4 ).

Find ( u' ) and ( v' ), then apply the product rule:

[ u' = 2(3x - \frac{4}{x}) \left(3 - \frac{-4}{x^2}\right) ] [ v' = 4(1 - x + 7x^2)^3(-1 + 14x) ]

Now, apply the product rule:

[ \frac{d}{dx}[(3x - \frac{4}{x})^2 (1 - x + 7x^2)^4] = u'v + uv' ]

[ = (2(3x - \frac{4}{x}) \left(3 - \frac{-4}{x^2}\right))(1 - x + 7x^2)^4 + (3x - \frac{4}{x})^2(4(1 - x + 7x^2)^3(-1 + 14x)) ]

So, the derivative of ( (3x - \frac{4}{x})^2 (1 - x + 7x^2)^4 ) is the expression above.

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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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