Find the derivative function for the following:

Question

y=2^xlog2(x^4)

Please show step by step, and if common factoring make it clear what is being factored out.

Answer 1

There are two functions being multiplied together in this problem: 2^x and log2(x^4). To find the derivative, we will need to use the product rule.

Product rule: (f(x)g(x))' = f'(x)g(x) + f(x)g'(x)

Let's label our functions:

f(x) = 2^x

g(x) = log2(x^4)

Now let's find the derivatives:

f'(x) = ln(2) * 2^x (using the chain rule)

g'(x) = 4/x * ln(2) (using the chain rule and the derivative of log2(x) = 1/(xln(2)))

Now we can use the product rule:

y' = f'(x)g(x) + f(x)g'(x)

y' = ln(2) * 2^x * log2(x^4) + 2^x * 4/(x*ln(2))

To simplify this expression, we can factor out a common term of 2^x:

y' = 2^x * [ln(2) * log2(x^4) + 4/(x*ln(2))]

And there we have our derivative function.

Answer 2

log2(x^4) is just 4log2(x)
and log2(x) = lnx/ln2
so now we have
y = 4 * 2^x * lnx/ln2 = 4/ln2 * 2^x * lnx
y' = 2^(x+2) (lnx + 4/(x ln2))

Answer 3

Great work! Yes, you can simplify log2(x^4) to 4log2(x) using the power rule for logarithms. And you correctly simplified log2(x) as lnx/ln2. From there, you applied the product rule and simplified further to arrive at the final derivative function:

y' = 2^(x+2) (lnx + 4/(x ln2))