Find the derivative of the function. Simplify and express the answer using only positive exponents only.
y = (x - 4)^2 (x + 3)
y= uv
y'=uv'+vu'
u=(x-4)^2 u'=2(x-4)
v=x+3 v'=1
-3^-x
2x^2
To find the derivative of the function y = (x - 4)^2 (x + 3), we can use the product rule and the chain rule.
The product rule states that if you have two functions f(x) and g(x), the derivative of their product is given by:
(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)
Let's apply this rule to the given function.
First, we differentiate the first factor (x - 4)^2 using the chain rule. The chain rule states that if you have a composite function, such as (f(g(x))), the derivative is given by:
(f(g(x)))' = f'(g(x)) * g'(x)
In this case, the composite function is (x - 4)^2, and its derivative is:
(2(x - 4)) * (1) = 2(x - 4)
Next, we differentiate the second factor (x + 3) which is just a linear function, so its derivative is 1.
Now we can apply the product rule:
y' = (x - 4)^2 * 1 + (x + 3) * 2(x - 4)
Simplifying this equation, we get:
y' = (x - 4)^2 + 2(x + 3)(x - 4)
Expanding the quadratic term, we have:
y' = (x^2 - 8x + 16) + 2(x^2 - 4x + 3x - 12)
Combining like terms, we get:
y' = x^2 - 8x + 16 + 2x^2 - 8x + 6x - 24
Simplifying further, we have:
y' = 3x^2 - 10x - 8
Therefore, the derivative of the function y = (x - 4)^2 (x + 3), simplified and expressed using only positive exponents, is y' = 3x^2 - 10x - 8.