What is the derivative of (1+4x)^5(3+x-x^2)^8?
1 Answer
Explanation:
You can differentiate this function by using the chain rule and the product rule.
Notice that your function can be written as
y = f(x) * g(x)
which means that its derivative can be determined using the product rule
d/dx(y) = [d/dx(f(x))] * g(x) + f(x) * d/dx(g(x))
In your case,
Differentiate these functions separately by using the chain rule. For
f(u) = u^5 , withu = (1 + 4x)
This will get you
d/dx(u^5) = d/(du)(u^5) * d/dx(u)
d/dx(u^5) = 5u^4 * d/dx(1+4x)
d/dx(1+4x)^5 = 5 * (1 + 4x)^4 * 4
For #g(x) you have
g(u_1) = u_1^8 , withu_1 = 3 + x - x^2
This will get you
d/dx(u_1^8) = d/(du_1)(u_1^8) * d/dx(u_1)
d/dx(u_1^8) = 8u_1^7 * d/dx(3 + x - x^2)
d/dx(3 + x - x^2)^8 = 8(3 + x - x^2)^7 * (1 - 2x)
Take these derivatives to your target calculation to get
y^' = 20 * (1 + 4x)^4 * (3 + x - x^2)^8 + (1 + 4x)^5 * 8 * (1-2x) * (3 + x - x^2)^7
y^' = 4(1 + 4x)^4 * (3 + x - x^2)^7 * [ 5 * (3 + x - x^2) + 2 (1-2x)(1 + 4x)]
This is equivalent to
y^' = 4(1 + 4x)^4 * (3 + x - x^2)^7 * [15 + 5x + 5x^2 + 2(1 + 2x - 8x^2)]
y^' = 4(1 + 4x)^4 * (3 + x - x^2)^7 * (15 + 5x - 5x^2 + 2 + 4x - 16x^2)
y^' = color(green)(4(1 + 4x)^4 * (3 + x - x^2)^7 * (-21x^2 + 9x + 17))