derivate of 2sin(x)-cos(x)-cot(x)
To find the derivative of the function f(x) = 2sin(x) - cos(x) - cot(x), we'll use the basic rules of differentiation. Let's go step by step:
Step 1: Differentiate the term 2sin(x):
The derivative of sin(x) with respect to x is cos(x), and since 2 is a constant, the derivative of 2sin(x) with respect to x is 2cos(x).
Step 2: Differentiate the term -cos(x):
The derivative of cos(x) with respect to x is -sin(x), and since -1 is a constant, the derivative of -cos(x) with respect to x is -(-1)sin(x) = sin(x).
Step 3: Differentiate the term -cot(x):
To differentiate cot(x), we need to simplify it first. cot(x) is equivalent to cos(x)/sin(x). Now, let's differentiate it using the quotient rule.
The quotient rule states that if u = f(x)/g(x), then the derivative of u with respect to x is (f'(x)g(x) - f(x)g'(x)) / (g(x))^2.
Let's apply this rule:
f(x) = cos(x) (numerator function)
g(x) = sin(x) (denominator function)
Now, we need to differentiate f(x) and g(x) individually:
f'(x) = -sin(x) (derivative of cos(x) is -sin(x))
g'(x) = cos(x) (derivative of sin(x) is cos(x))
Using the quotient rule, the derivative of -cot(x) is:
(-sin(x) * sin(x) - cos(x) * cos(x)) / (sin(x))^2
Simplifying further, we get:
-(sin^2(x) + cos^2(x)) / sin^2(x) = -1 / sin^2(x)
Step 4: Combine the derivatives:
Now that we have the individual derivatives, we can add them up:
2cos(x) + sin(x) - 1 / sin^2(x)
Therefore, the derivative of f(x) = 2sin(x) - cos(x) - cot(x) is:
2cos(x) + sin(x) - 1 / sin^2(x)