How do you differentiate #f(x)=arctanx*3x# using the product rule?

Answer 1

#d/dxf(x) = (3x)/(1+x^2) + 3arctanx#

#" "# Differentiating #f(x)# is determined by applying product differentiation. #" "# Product Differentiation: #" "# #color(blue)(d/dx(u(x)xxv(x))=d/dxu(x)xxv(x) + u(x)xxd/dxv(x))# #" "# #" "# #d/dxf(x) = d/dx((arctanx)xx3x)# #" "# #d/dxf(x) = d/dx(arctanx)xx3x + arctanx xx d/dx(3x)# #" "# #d/dxf(x) = 1/(1+x^2)xx3x + arctanx xx 3# #" "# #d/dxf(x) = (3x)/(1+x^2) + 3arctanx#
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Answer 2

To differentiate the function (f(x) = \arctan(x) \cdot 3x) using the product rule, we apply the formula: ((uv)' = u'v + uv'), where (u = \arctan(x)) and (v = 3x).

First, we find the derivatives of (u) and (v):

(u' = \frac{1}{1 + x^2}) (derivative of (\arctan(x)) with respect to (x))

(v' = 3) (derivative of (3x) with respect to (x))

Then, we substitute these derivatives into the product rule formula:

(f'(x) = u'v + uv' = \frac{1}{1 + x^2} \cdot 3x + \arctan(x) \cdot 3)

Thus, the derivative of (f(x) = \arctan(x) \cdot 3x) with respect to (x) using the product rule is (f'(x) = \frac{3x}{1 + x^2} + 3\arctan(x)).

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Answer from HIX Tutor

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.

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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