# How do you find the derivative of #x^3*arctan(7x)#?

Firstly let's differentiate this function using implicit and logarithmic differentiation:

I can also give you an alternative way of finding this derivative, using the product rule...

This means that:

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To find the derivative of (x^3 \cdot \arctan(7x)), you can use the product rule of differentiation. The product rule states that if (u(x)) and (v(x)) are differentiable functions of (x), then the derivative of their product is given by ((u \cdot v)' = u'v + uv').

Let (u(x) = x^3) and (v(x) = \arctan(7x)).

Now, (u'(x) = 3x^2) (derivative of (x^3)) and (v'(x) = \frac{7}{1 + (7x)^2}) (derivative of (\arctan(7x)) using the chain rule).

Applying the product rule:

[ \begin{align*} (uv)' &= u'v + uv' \ &= (3x^2) \cdot \arctan(7x) + x^3 \cdot \frac{7}{1 + (7x)^2} \ &= 3x^2 \cdot \arctan(7x) + \frac{7x^3}{1 + 49x^2}. \end{align*} ]

So, the derivative of (x^3 \cdot \arctan(7x)) is (3x^2 \cdot \arctan(7x) + \frac{7x^3}{1 + 49x^2}).

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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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