How do you use the Product Rule to find the derivative of #g(x) = (2x^2 + 4x - 3) ( 5x^3 + 2x + 2)#?

Answer 1

#g^' = 50x^4 + 80x^3 - 33x^2 + 24x + 2#

Notice that your function can be written as the product of two other functions

#g(x) = underbrace((2x^2 + 4x - 3))_(color(orange)(f(x))) * underbrace((5x^3 + 2x + 2))_(color(purple)(h(x)))#

This means that you can rule the product rule to get

#color(blue)(d/dx(g(x)) = [d/dx(f(x))] * h(x) + f(x) * d/dx(h(x)))#

Using this formula will get you

#d/dx(g(x)) = [d/dx(2x^2 + 4x - 3)] * (5x^3 + 2x + 2) + (2x^2 + 4x - 3) * d/dx(5x^3 + 2x + 2)#
#g^' = (4x + 4) * (5x^3 + 2x + 2) + (2x^2 + 4x - 3) * (15x^2 + 2)#
#g^' = 20x^4 + 8x^2 + 8x + 20x^3 + 8x + 8 + 30x^4 + 4x^2 + 60x^3 + 8x -45x^2 - 6#
#g^' = color(green)(50x^4 + 80x^3 - 33x^2 + 24x + 2)#
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Answer 2

To use the Product Rule to find the derivative of ( g(x) = (2x^2 + 4x - 3)(5x^3 + 2x + 2) ), follow these steps:

The Product Rule states that if you have two functions ( u(x) ) and ( v(x) ), the derivative of their product ( u(x)v(x) ) is given by: [ \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) ]

Let's identify ( u(x) ) and ( v(x) ) in the given function: [ u(x) = 2x^2 + 4x - 3 ] [ v(x) = 5x^3 + 2x + 2 ]

First, find the derivatives of ( u(x) ) and ( v(x) ): [ u'(x) = \frac{d}{dx}(2x^2 + 4x - 3) = 4x + 4 ] [ v'(x) = \frac{d}{dx}(5x^3 + 2x + 2) = 15x^2 + 2 ]

Now, apply the Product Rule: [ g'(x) = u'(x)v(x) + u(x)v'(x) ] [ g'(x) = (4x + 4)(5x^3 + 2x + 2) + (2x^2 + 4x - 3)(15x^2 + 2) ]

To simplify, distribute and combine like terms: [ g'(x) = (4x + 4)(5x^3 + 2x + 2) + (2x^2 + 4x - 3)(15x^2 + 2) ] [ g'(x) = 20x^4 + 8x^2 + 8x + 16 + 30x^4 + 4x^2 + 8x^2 + 16x - 45x^2 - 6 ] [ g'(x) = 50x^4 + 25x^2 + 24x + 10 ]

Therefore, the derivative of ( g(x) = (2x^2 + 4x - 3)(5x^3 + 2x + 2) ) is ( g'(x) = 50x^4 + 25x^2 + 24x + 10 ).

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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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