f(x) (x^2+lnx)(2+e^2)
derivative
also,
f(x)= (3x+x^4)^(3/2)/(7x^2-1)
for the first one, since 2+e^2 is a constant, you simply get
f'(x) = (2+e^2)(2x + 1/x)
(are you sure there was no typo ?)
for the 2nd use the quotient rule, give it a try, let me know what you got
the first one should be (x^2+lnx)(2+e^x) sorry
f(x) = (x^2+lnx)(2+e^x)
f'(x) = (x^2 + lnx)(e^x) + (2+e^x)(2x + 1/x)
I used the product rule,
expand and simplify if needed
To find the derivative of the given functions, we can use the rules of differentiation. Let's start with the first function:
f(x) = (x^2 + ln(x))(2 + e^2)
To find the derivative, we'll use the product rule, which states that if we have two functions multiplied together, the derivative will be the derivative of the first function times the second function, plus the first function times the derivative of the second function.
Let's apply the product rule to f(x):
f'(x) = (x^2 + ln(x)) * derivative of (2 + e^2) + (2 + e^2) * derivative of (x^2 + ln(x))
Next, we need to find the derivative of each of the individual functions involved. Taking derivatives:
Derivative of (2 + e^2) is 0, because it is the derivative of a constant term.
Derivative of (x^2 + ln(x)) can be found separately for each term:
- Derivative of x^2 is 2x.
- Derivative of ln(x) is 1/x.
So, substituting these results back into our derivative expression:
f'(x) = (x^2 + ln(x)) * 0 + (2 + e^2) * (2x + 1/x)
Simplifying further:
f'(x) = (2 + e^2)(2x + 1/x)
Therefore, the derivative of f(x) = (x^2 + ln(x))(2 + e^2) is f'(x) = (2 + e^2)(2x + 1/x).
Now, let's find the derivative of the second function:
f(x) = (3x + x^4)^(3/2) / (7x^2 - 1)
To differentiate this, we'll use the quotient rule, which states that if we have a function in the form of f(x) = g(x) / h(x), the derivative will be (h(x) * g'(x) - g(x) * h'(x)) / (h(x))^2.
In our case, g(x) = (3x + x^4)^(3/2) and h(x) = (7x^2 - 1).
Now let's calculate the derivatives:
- Derivative of g(x) = (3/2) * (3x + x^4)^(1/2) * (3 + 4x^3) using the chain rule.
- Derivative of h(x) = 14x.
Applying the quotient rule formula:
f'(x) = [(7x^2 - 1) * (3/2) * (3x + x^4)^(1/2) * (3 + 4x^3) - (3x + x^4)^(3/2) * 14x] / (7x^2 - 1)^2
After simplifying and factoring, you will have the derivative of f(x).
It is important to note that the calculations might involve further simplification or combining terms, so it's always a good idea to double-check your work and simplify the final expression if possible.