# How do you differentiate # arctan(8^x)#?

This kind of problem is usually done with implicit differentiation. We begin by writing the function in terms of

Next, we use the definition of

We can now proceed with the implicit differentiation with respect to

In line [B] we note that the definition of

The issue with this is the derivative still contains a term of

Note how the picture illustrates the fact that

Using trigonometry, we can now evaluate what the value of

Thus:

Finally:

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To differentiate arctan(8^x) with respect to x, use the chain rule. The derivative is (1/(1 + (8^x)^2)) * (d/dx)(8^x). Apply the derivative of 8^x, which is (8^x) * ln(8). So, the final result is (1/(1 + 64^x)) * (8^x) * ln(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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