log(x + 4) ----------*cot(7*x) log(2)
(log(x + 4)/log(2))*cot(7*x)
Apply the quotient rule, which is:
and .
To find :
Apply the product rule:
; to find :
There are multiple ways to do this derivative.
Rewrite the function to be differentiated:
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Rewrite the function to be differentiated:
Apply the quotient rule, which is:
and .
To find :
Let .
The derivative of sine is cosine:
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 :
Let .
The derivative of cosine is negative sine:
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:
Now plug in to the quotient rule:
The result of the chain rule is:
Rewrite the function to be differentiated:
Apply the quotient rule, which is:
and .
To find :
Let .
The derivative of cosine is negative sine:
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 :
Let .
The derivative of sine is cosine:
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:
Now plug in to the quotient rule:
; to find :
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
Differentiate term by term:
Apply the power rule: goes to
The derivative of the constant is zero.
The result is:
The result of the chain rule is:
The result is:
To find :
The derivative of the constant is zero.
Now plug in to the quotient rule:
Now simplify:
The answer is:
/ 2 \ cot(7*x) \-7 - 7*cot (7*x)/*log(x + 4) -------------- + ----------------------------- (x + 4)*log(2) log(2)
/ 2 \ cot(7*x) 14*\1 + cot (7*x)/ / 2 \ - -------- - ------------------ + 98*\1 + cot (7*x)/*cot(7*x)*log(4 + x) 2 4 + x (4 + x) ------------------------------------------------------------------------ log(2)
/ 2 \ / 2 \ 2*cot(7*x) 21*\1 + cot (7*x)/ / 2 \ / 2 \ 294*\1 + cot (7*x)/*cot(7*x) ---------- + ------------------ - 686*\1 + cot (7*x)/*\1 + 3*cot (7*x)/*log(4 + x) + ---------------------------- 3 2 4 + x (4 + x) (4 + x) ----------------------------------------------------------------------------------------------------------------- log(2)