Differentiate y=(5x-2)^3(3-x)^6 using the product rule.
i don't get the step where they find the common factor could you help me by doing it step by step please ?
thank you!
There is no such step that i have ever heard of.
Let
u (x) = (5x-2)^3 and
v (x) = (3-x)^6
You want the derivative of u*v with respect to x.
dy/dx = d(uv)/dx = u dv/dx + v du/dx
That is the product rule.
Now all you have to calculate is du/dx and dv/dx, multiply the v and u terms and add the result.
thanks!
there is an example here and where i get lost is a step where it goes from
dy/dx = d(uv)/dx = u dv/dx + v du/dx
= (5x-2)^3[-6(3-x)^5]+(3-x)^6[15(5x-2)^2]
= (5x-2)^2(3-x)^5(-30x+12+45-15x)
is there a step between there that you could show me how to do so i can understand it ?
thank you!
You must be talking about the simplication after you found the derivative.
look at the line ....
(5x-2)^3[-6(3-x)^5]+(3-x)^6[15(5x-2)^2]
= -6(5x-2)^3(3-x)^5 + 15(3-x)^6(5x-2)^2
now I see the following common factors
-3 for the constants
(5x-2)^2 and
(3-x)^5
so
= -3(5x-2)^2(3-x)^5[2(5x-2) - 5(3-x)]
= -3(5x-2)^2(3-x)^5[10x - 4 - 15 + 5x]
= -3(5x-2)^2(3-x)^5(15x - 19)
Your last line is not fully factored.
ah OK, now i get it.
Thank you so much!!
hey sorry, one more thing.
How do you find -3
as in
now I see the following common factors
-3
how ?
(-30x+12+45-15x)
=-3(15x-19)
Of course! I'd be happy to walk you through the steps.
To differentiate the given function, y = (5x - 2)^3(3 - x)^6, using the product rule, we need to find the derivatives of each factor separately and then apply the product rule.
Let's start by finding the derivative of the first factor, (5x - 2)^3. To do this, we can use the chain rule.
Step 1: Identify the inner function and its derivative.
The inner function here is (5x - 2), and its derivative is 5.
Step 2: Differentiate the outer function.
If our outer function is u^n, where n is a constant, the derivative is nu^(n-1).
In this case, the outer function is u^3, so its derivative is 3u^(3-1) = 3u^2.
Step 3: Apply the chain rule.
Multiply the derivative of the outer function by the derivative of the inner function.
So, the derivative of (5x - 2)^3 is 3(5x - 2)^2 * 5 = 15(5x - 2)^2.
Now, let's find the derivative of the second factor, (3 - x)^6, using the same steps.
Step 1: Identify the inner function and its derivative.
The inner function here is (3 - x), and its derivative is -1.
Step 2: Differentiate the outer function.
If our outer function is u^n, where n is a constant, the derivative is nu^(n-1).
In this case, the outer function is u^6, so its derivative is 6u^(6-1) = 6u^5.
Step 3: Apply the chain rule.
Multiply the derivative of the outer function by the derivative of the inner function.
So, the derivative of (3 - x)^6 is 6(3 - x)^5 * (-1) = -6(3 - x)^5.
Now that we have the derivatives of both factors, let's differentiate the entire function using the product rule.
The product rule states that if we have two functions u(x) and v(x), the derivative of their product, u(x)v(x), is given by:
(uv)' = u'v + uv'
To apply the product rule to our function y = (5x - 2)^3(3 - x)^6, we differentiate each factor and then apply the product rule:
dy/dx = [15(5x - 2)^2] * (3 - x)^6 + (5x - 2)^3 * [-6(3 - x)^5]
Simplifying this expression will give you the final answer.