Finding the Derivative of (1+4x)^(5)(3+x^(2))^(8)
This article will guide you through the process of finding the derivative of the function (1+4x)^(5)(3+x^(2))^(8).
Understanding the Concept:
We will be using the product rule and chain rule to find the derivative of this function.
Product Rule: The product rule states that the derivative of the product of two functions, u(x) and v(x), is:
d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
Chain Rule: The chain rule states that the derivative of a composite function, f(g(x)), is:
d/dx [f(g(x))] = f'(g(x)) * g'(x)
Applying the Rules:

Identify the two functions:
 u(x) = (1+4x)^(5)
 v(x) = (3+x^(2))^(8)

Find the derivatives of each function:
 u'(x): We'll use the chain rule.
 f(x) = x^(5) => f'(x) = 5x^(4)
 g(x) = 1+4x => g'(x) = 4
 u'(x) = f'(g(x)) * g'(x) = 5(1+4x)^(4) * 4 = 20(1+4x)^(4)
 v'(x): We'll use the chain rule again.
 f(x) = x^(8) => f'(x) = 8x^(7)
 g(x) = 3+x^(2) => g'(x) = 2x
 v'(x) = f'(g(x)) * g'(x) = 8(3+x^(2))^(7) * 2x = 16x(3+x^(2))^(7)
 u'(x): We'll use the chain rule.

Apply the product rule:
 d/dx [(1+4x)^(5)(3+x^(2))^(8)] = u'(x)v(x) + u(x)v'(x)
 = 20(1+4x)^(4) * (3+x^(2))^(8) + (1+4x)^(5) * 16x(3+x^(2))^(7)

Simplify the expression (optional):
 You can factor out common terms like (1+4x)^(4) and (3+x^(2))^(7) to make the expression cleaner.
Final Result:
The derivative of the function (1+4x)^(5)(3+x^(2))^(8) is:
20(1+4x)^(4)(3+x^(2))^(8) + 16x(1+4x)^(5)(3+x^(2))^(7)
Remember to always doublecheck your work and apply the rules carefully.