Find the derivative function for the following:
y=2^xlog2(x^4)
Please show step by step, and if common factoring make it clear what is being factored out.
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.
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))
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))
To find the derivative of the function y = 2^x * log2(x^4), we can use the product rule and the chain rule. Here are the step-by-step calculations:
Step 1: Apply the product rule to the given function y = u * v, where u = 2^x and v = log2(x^4).
The product rule states that (u * v)' = u' * v + u * v', where u' and v' are the derivatives of u and v, respectively.
So, let's find the derivatives of u and v.
Step 2: Find the derivative of u = 2^x.
To do this, we use the chain rule. The chain rule states that if we have a function h(g(x)), then the derivative is given by h'(g(x)) * g'(x).
The derivative of 2^x can be found as follows:
Let h(t) = 2^t, where t = x.
Then, using the chain rule, h'(t) = d(2^t)/dt = (ln(2)) * (2^t).
Now, substitute back x for t to get the derivative of u:
u' = h'(x) = (ln(2)) * (2^x).
Step 3: Find the derivative of v = log2(x^4).
The derivative of log2(x^4) can be found using the chain rule as well. We let g(t) = log2(t), where t = x^4.
Then, using the chain rule, g'(t) = 1 / (t * ln(2)).
Now, substitute back x^4 for t to get the derivative of v:
v' = g'(x^4) = 1 / (x^4 * ln(2)).
Step 4: Now that we have u' and v', we can apply the product rule to find the derivative of y = u * v.
Using the product rule, (u * v)' = u' * v + u * v'.
Substituting the values we found in Steps 2 and 3:
(u * v)' = (ln(2))(2^x)(1 / (x^4 * ln(2))) + (2^x)(1 / (x^4 * ln(2))).
Simplifying the expression, we have:
(u * v)' = (ln(2))(2^x) / (x^4 * ln(2)) + (2^x) / (x^4 * ln(2)).
Further simplification can be done by factoring out a common term of (2^x) / (x^4 * ln(2)):
(u * v)' = (2^x / (x^4 * ln(2))) * ((ln(2)) + 1).
Therefore, the derivative function is:
y' = (2^x / (x^4 * ln(2))) * ((ln(2)) + 1).
That's how you find the derivative of the function y = 2^xlog2(x^4) step-by-step using the product rule and the chain rule.