find the derivative of 4x(lnx+ln9)-9x+e
To find the derivative of the given function, we can use the rules of differentiation.
The function consists of several terms, so we'll find the derivative of each term separately and then add them up.
1. Derivative of 4x(lnx+ln9):
To find the derivative of x times the sum of two natural logarithms, we use the product rule. The product rule states that the derivative of the product of two functions is equal to the derivative of the first function times the second function plus the first function times the derivative of the second function.
The first function is x, and the second function is (lnx + ln9).
The derivative of the first function (x) is 1.
The derivative of the second function (lnx + ln9) is 1/x because the derivative of ln(x) is equal to 1/x.
Therefore, the derivative of 4x(lnx + ln9) is:
(4)(1)(lnx + ln9) + (4)(1/x)(x)
= 4(lnx + ln9) + 4
2. Derivative of -9x:
The derivative of -9x is simply -9.
3. Derivative of e:
The derivative of e, where e is a constant, is zero since e is a constant value and doesn't change with respect to x.
To find the derivative of the whole function, we sum up the derivatives of each term:
4(lnx + ln9) + 4 - 9 + 0
= 4lnx + 4ln9 - 5
So, the derivative of 4x(lnx+ln9) - 9x + e is 4lnx + 4ln9 - 5.