/ 4\ log\3*x / --------- 3 7*x - 9
/ / 4\\ d |log\3*x /| --|---------| dx| 3 | \ 7*x - 9/
Apply the quotient rule, which is:
and .
To find :
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The result of the chain rule is:
To find :
Differentiate term by term:
The derivative of the constant is zero.
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The result is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
2 / 4\ 4 21*x *log\3*x / ------------ - --------------- / 3 \ 2 x*\7*x - 9/ / 3 \ \7*x - 9/
/ / 3 \ \ | | 21*x | / 4\| | 21*x*|-1 + ---------|*log\3*x /| | | 3| | | 2 84*x \ -9 + 7*x / | 2*|- -- - --------- + -------------------------------| | 2 3 3 | \ x -9 + 7*x -9 + 7*x / ------------------------------------------------------ 3 -9 + 7*x
/ / 3 6 \ \ | / 3 \ | 126*x 1323*x | / 4\| | | 21*x | 21*|1 - --------- + ------------|*log\3*x /| | 252*|-1 + ---------| | 3 2| | | | 3| | -9 + 7*x / 3\ | | |4 126 \ -9 + 7*x / \ \-9 + 7*x / / | 2*|-- + --------- + -------------------- - -------------------------------------------| | 3 3 3 3 | \x -9 + 7*x -9 + 7*x -9 + 7*x / --------------------------------------------------------------------------------------- 3 -9 + 7*x