Proving the Derivative of csc(x) * csc(x) * cot(x)
To prove that the derivative of the function y = csc(x) * csc(x) * cot(x) is computed correctly, we will use the product rule and the chain rule of differentiation. The derivative of the product of multiple functions can be calculated using the general product rule.
Let us denote:
- u = csc(x)
- v = csc(x)
- w = cot(x)
Then, we have:
y = u * v * w = (csc(x))^2 * cot(x)
Using the Product Rule
We will find the derivative y’ as follows:
y’ = rac{d}{dx}[u * v * w] = u’ * v * w + u * v’ * w + u * v * w’
Calculating the Derivatives
Now we need to calculate the derivatives of u, v, and w:
- u’ = -csc(x) * cot(x)
- v’ = -csc(x) * cot(x)
- w’ = -csc^2(x)
Substituting Derivatives into the Product Rule
Now we can substitute back into the product rule:
y’ = (-csc(x) * cot(x)) * csc(x) * cot(x) + csc(x) * (-csc(x) * cot(x)) * cot(x) + csc(x) * csc(x) * (-csc^2(x))
Simplifying the Expression
Now let’s simplify each term:
- First term: -csc^2(x) * cot^2(x)
- Second term: -csc^2(x) * cot^2(x)
- Third term: -csc^4(x)
The combined expression becomes:
y’ = -2csc^2(x) * cot^2(x) – csc^4(x)
Final Result
Therefore, we have proved that the derivative of the function csc(x) * csc(x) * cot(x) is given by:
y’ = -2csc^2(x) * cot^2(x) – csc^4(x)
This concludes our proof.