Question

F(x)=cot(x)tan(x) use product property to find derivative

Answer 1

To find the derivative of f(x) = cot(x)tan(x) using the product rule, differentiate each function separately and apply the quotient rule. Then, use the product rule to find the derivative of f(x).

Step-by-step explanation:

To find the derivative of f(x) = cot(x)tan(x) using the product rule, we can write f(x) as the product of two functions: f(x) = cot(x) * tan(x).

Let's differentiate each function separately:

1. Differentiating cot(x):
   1. cot(x) = cos(x) / sin(x)
   2. Differentiating cos(x) gives us -sin(x)
   3. Differentiating sin(x) gives us cos(x)
   4. Using the quotient rule, we get the derivative of cot(x) as -sin(x) / sin^2(x)
2. Differentiating tan(x):
   1. Tan(x) = sin(x) / cos(x)
   2. Differentiating sin(x) gives us cos(x)
   3. Differentiating cos(x) gives us -sin(x)
   4. Using the quotient rule, we get the derivative of tan(x) as (cos(x) * cos(x)) / cos^2(x)

Now, we can use the product rule to find the derivative of f(x):

f'(x) = cot(x) * (cos(x) * cos(x)) / cos^2(x) - (sin(x) / sin^2(x)) * tan(x)

From here, you can simplify the expression if needed.