Find the derivative of y=x(2^(4x)).
To find the derivative of the function y = x(2^(4x)), we can use the product rule and the chain rule.
The product rule states that if we have a function in the form f(x)g(x), the derivative of that function is f'(x)g(x) + f(x)g'(x).
The chain rule states that if we have a composition of functions, f(g(x)), the derivative of that composition is f'(g(x)) * g'(x).
So, let's apply these rules to find the derivative of y = x(2^(4x)):
Step 1: Identify the two functions involved in the product rule.
In this case, f(x) = x and g(x) = 2^(4x).
Step 2: Find the derivatives of f(x) and g(x).
The derivative of f(x) = x is f'(x) = 1 (since the derivative of x with respect to x is 1).
The derivative of g(x) = 2^(4x) requires the chain rule. Let's define h(x) = 4x and apply the chain rule:
h'(x) = 4 (since the derivative of 4x with respect to x is 4).
Now, using the chain rule, the derivative of g(x) = 2^(4x) is g'(x) = ln(2) * 2^(4x) * h'(x) = 4ln(2) * 2^(4x).
Step 3: Apply the product rule.
Using the product rule, the derivative of y = f'(x)g(x) + f(x)g'(x) becomes:
y' = 1 * 2^(4x) + x * 4ln(2) * 2^(4x).
Simplifying, we get:
y' = 2^(4x) + 4ln(2) * x * 2^(4x).
Therefore, the derivative of y = x(2^(4x)) is y' = 2^(4x) + 4ln(2) * x * 2^(4x).