# How do you differentiate #y=tan^-1(2x^4)#?

Substitute into (A) and change u back to x

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To differentiate (y = \tan^{-1}(2x^4)) with respect to (x), you use the chain rule. The derivative of (\tan^{-1}u) with respect to (u) is (\frac{1}{1+u^2}), and then you multiply by the derivative of (u) with respect to (x), where (u = 2x^4).

The derivative of (u = 2x^4) with respect to (x) is (du/dx = 8x^3).

So, the derivative of (y) with respect to (x) is: [ \frac{dy}{dx} = \frac{1}{1+(2x^4)^2} \cdot 8x^3 = \frac{8x^3}{1+4x^8}. ]

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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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