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How do you differentiate #(5-x^2)(x^3-3x+3)#?

Answer 1

Ezekiel Alexander

Just multiply it out and then use the Power Rule.

#= 5x^3 - 15x + 15 - x^5 + 3x^3 - 3x^2#

#f(x) = - x^5 + 8x^3 - 3x^2 - 15x + 15#

#d/(dx)[f(x)] = -5x^4 + 24x^2 - 6x - 15#

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Answer 2

Isabella Ackerman

To differentiate the expression (5-x^2)(x^3-3x+3), you can use the product rule, which states that if you have two functions u(x) and v(x), then the derivative of their product is given by:

(uv)' = u'v + uv'

Applying this rule to the given expression:

(5 - x^2)'(x^3 - 3x + 3) + (5 - x^2)(x^3 - 3x + 3)'

Differentiating each term separately:

(5 - x^2)' = -2x (x^3 - 3x + 3)' = 3x^2 - 3

Now, plug these derivatives back into the product rule:

(-2x)(x^3 - 3x + 3) + (5 - x^2)(3x^2 - 3)

Expanding and simplifying:

-2x^4 + 6x^2 - 6x + 15x^2 - 3x^2 + 3x^2

Combining like terms:

-2x^4 + 21x^2 - 6x

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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.