/ /2*x - 3\\ sin|log|-------|| \ \ x + 4 //
sin(log((2*x - 3)/(x + 4)))
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
Apply the quotient rule, which is:
and .
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:
To find :
Differentiate term by term:
The derivative of the constant is zero.
Apply the power rule: goes to
The result is:
Now plug in to the quotient rule:
The result of the chain rule is:
The result of the chain rule is:
Now simplify:
The answer is:
/ 2 2*x - 3 \ / /2*x - 3\\ (x + 4)*|----- - --------|*cos|log|-------|| |x + 4 2| \ \ x + 4 // \ (x + 4) / -------------------------------------------- 2*x - 3
/ / /-3 + 2*x\\ / /-3 + 2*x\\ / -3 + 2*x\ / /-3 + 2*x\\\ | cos|log|--------|| 2*cos|log|--------|| |2 - --------|*sin|log|--------||| / -3 + 2*x\ | \ \ 4 + x // \ \ 4 + x // \ 4 + x / \ \ 4 + x //| |2 - --------|*|- ------------------ - -------------------- - ---------------------------------| \ 4 + x / \ 4 + x -3 + 2*x -3 + 2*x / ------------------------------------------------------------------------------------------------ -3 + 2*x
/ 2 \ | / /-3 + 2*x\\ / /-3 + 2*x\\ / -3 + 2*x\ / /-3 + 2*x\\ / /-3 + 2*x\\ / -3 + 2*x\ / /-3 + 2*x\\ / -3 + 2*x\ / /-3 + 2*x\\| |2*cos|log|--------|| 8*cos|log|--------|| |2 - --------| *cos|log|--------|| 4*cos|log|--------|| 6*|2 - --------|*sin|log|--------|| 3*|2 - --------|*sin|log|--------||| / -3 + 2*x\ | \ \ 4 + x // \ \ 4 + x // \ 4 + x / \ \ 4 + x // \ \ 4 + x // \ 4 + x / \ \ 4 + x // \ 4 + x / \ \ 4 + x //| |2 - --------|*|-------------------- + -------------------- - ---------------------------------- + -------------------- + ----------------------------------- + -----------------------------------| \ 4 + x / | 2 2 2 (-3 + 2*x)*(4 + x) 2 (-3 + 2*x)*(4 + x) | \ (4 + x) (-3 + 2*x) (-3 + 2*x) (-3 + 2*x) / ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -3 + 2*x