cot(5*x) + log(x)
cot(5*x) + log(x)
Differentiate term by term:
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:
The derivative of is .
The result is:
Now simplify:
The answer is:
1 2 -5 + - - 5*cot (5*x) x
1 / 2 \ - -- + 50*\1 + cot (5*x)/*cot(5*x) 2 x
/ 2 \ |1 / 2 \ 2 / 2 \| 2*|-- - 125*\1 + cot (5*x)/ - 250*cot (5*x)*\1 + cot (5*x)/| | 3 | \x /