How do you use the Product Rule to find the derivative of #g(x) = (2x^2 + 4x - 3) ( 5x^3 + 2x + 2)#?
Notice that your function can be written as the product of two other functions
This means that you can rule the product rule to get
Using this formula will get you
By signing up, you agree to our Terms of Service and Privacy Policy
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 ).
By signing up, you agree to our Terms of Service and Privacy Policy
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.
- How do you use the chain rule to differentiate #f(x)=37-sec^3(2x)#?
- How do you integrate #f(x)=3# using the power rule?
- If #f(x) =-e^(-3x-7) # and #g(x) = (lnx)^2 #, what is #f'(g(x)) #?
- How do you differentiate # y = 4(x-(3-5x)^4)^2# using the chain rule?
- How do you differentiate # y= sqrt((3x)/(2x-3))# using the chain rule?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7