log(3)*(4*x + 5) ---------------- / ___\ 2*cot\\/ x /
(log(3)*(4*x + 5))/((2*cot(sqrt(x))))
Apply the quotient rule, which is:
and .
To find :
The derivative of a constant times a function is the constant times the derivative of the function.
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:
So, the result is:
To find :
The derivative of a constant times a function is the constant times the derivative of the function.
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 :
Apply the power rule: goes to
The result of the chain rule is:
To find :
Let .
The derivative of cosine is negative sine:
Then, apply the chain rule. Multiply by :
Apply the power rule: goes to
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 :
Apply the power rule: goes to
The result of the chain rule is:
To find :
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
Apply the power rule: goes to
The result of the chain rule is:
Now plug in to the quotient rule:
So, the result is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
/ 2/ ___\\ 1 \-1 - cot \\/ x //*(4*x + 5)*log(3) 4*------------*log(3) - ----------------------------------- / ___\ ___ 2/ ___\ 2*cot\\/ x / 4*\/ x *cot \\/ x /
/ / / 2/ ___\\\\ | |2 1 2*\1 + cot \\/ x //|| | (5 + 4*x)*|- + --------------- - -------------------|| | |x 3/2 / ___\ 2/ ___\ || / 2/ ___\\ | 2 \ x *cot\\/ x / x*cot \\/ x / /| \1 + cot \\/ x //*|---------------- - -----------------------------------------------------|*log(3) | ___ / ___\ 8 | \\/ x *cot\\/ x / / --------------------------------------------------------------------------------------------------- / ___\ cot\\/ x /
/ / / 2/ ___\\\\ | |2 1 2*\1 + cot \\/ x //|| | / 2\ 24*|- + --------------- - -------------------|| | | / 2/ ___\\ / 2/ ___\\ / 2/ ___\\ | |x 3/2 / ___\ 2/ ___\ || / 2/ ___\\ | | 4 3 6 10*\1 + cot \\/ x // 6*\1 + cot \\/ x // 6*\1 + cot \\/ x // | \ x *cot\\/ x / x*cot \\/ x / /| \1 + cot \\/ x //*|(5 + 4*x)*|---- + ---------------- + ------------- - -------------------- - ------------------- + --------------------| - ----------------------------------------------|*log(3) | | 3/2 5/2 2/ ___\ 2 / ___\ 3/2 2/ ___\ 2 3/ ___\ 3/2 4/ ___\ | / ___\ | \ \x x *cot \\/ x / x *cot\\/ x / x *cot \\/ x / x *cot \\/ x / x *cot \\/ x / / cot\\/ x / / --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 16