The derivative of csc^2x is -2.csc^2(x).cot(x)
How to calculate the derivative of csc^2x
Note that in this post we will be looking at differentiating csc2(x) which is not the same as differentiating csc(2x). Here is our post dealing with how to differentiate csc(2x).
There are two methods that can be used for calculating the derivative of cosec^2x.
The first method is by using the product rule for derivatives (since csc2(x) can be written as csc(x).csc(x)).
The second method is by using the chain rule for differentiation.
Finding the derivative of csc^2x using the product rule
The product rule for differentiation states that the derivative of f(x).g(x) is f’(x)g(x) + f(x).g’(x)
The Product Rule:
For two differentiable functions f(x) and g(x)
If F(x) = f(x).g(x)
Then the derivative of F(x) is F'(x) = f’(x)g(x) + f(x)g'(x)
First, let F(x) = csc2(x)
Then remember that csc2(x) is equal to csc(x).csc(x)
So F(x) = csc(x)csc(x)
By setting f(x) and g(x) as csc(x) means that F(x) = f(x).g(x) and we can apply the product rule to find F'(x)
| F'(x) | = f'(x)g(x) + f(x)g'(x) | Product Rule Definition |
| = f'(x)csc(x) + csc(x)g'(x) | f(x) = g(x) = csc(x) | |
| = -csc(x)cot(x)csc(x) + csc(x)(-csc(x)cot(x)) | f'(x) = g(‘x) = -csc(x)cot(x) | |
| = -2csc2(x)cot(x) |
Using the product rule, the derivative of csc^2x is -2csc^2(x)cot(x)
Finding the derivative of csc^2x using the chain rule
The chain rule is useful for finding the derivative of a function which could have been differentiated had it been in x, but it is in the form of another expression which could also be differentiated if it stood on its own.
In this case:
- We know how to differentiate csc(x) (the answer is -csc(x)cot(x))
- We know how to differentiate x2 (the answer is 2x)
This means the chain rule will allow us to perform the differentiation of the expression csc^2x.
Using the chain rule to find the derivative of csc^2x
Although the expression csc2x contains no parenthesis, we can still view it as a composite function (a function of a function).
We can write csc2x as (csc(x))2.
Now the function is in the form of x2, except it does not have x as the base, instead it has another function of x (csc(x)) as the base.
Let’s call the function of the base g(x), which means:
g(x) = csc(x)
From this it follows that:
csc(x)2 = g(x)2
So if the function f(x) = x2 and the function g(x) = csc(x), then the function (csc(x))2 can be written as a composite function.
f(x) = x2
f(g(x)) = g(x)2 (but g(x) = csc(x))
f(g(x)) = (csc(x))2
Let’s define this composite function as F(x):
F(x) = f(g(x)) = (csc(x))2
We can find the derivative of csc^2x (F'(x)) by making use of the chain rule.
The Chain Rule:
For two differentiable functions f(x) and g(x)
If F(x) = f(g(x))
Then the derivative of F(x) is F'(x) = f’(g(x)).g’(x)
Now we can just plug f(x) and g(x) into the chain rule.
How to find the derivative of csc^2x using the Chain Rule:
| F'(x) | = f'(g(x)).g'(x) | Chain Rule Definition |
| = f'(g(x))(-csc(x)cot(x)) | g(x) = csc(x) ⇒ g'(x) = -csc(x)cot(x) | |
| = (2csc(x)).(csc(x)tan(x)) | f(g(x)) = (csc(x))2 ⇒ f'(g(x)) = 2csc(x) | |
| = -2csc2(x)cot(x) |
Using the chain rule, the derivative of csc^2x is -2.csc^2(x).cot(x)
Finally, just a note on syntax and notation: csc^2x is sometimes written in the forms below (with the derivative as per the calculations above). Just be aware that not all of the forms below are mathematically correct.
| csc2x | ► Derivative of csc2x = -2csc2(x)cot(x) |
| csc^2(x) | ► Derivative of csc^2(x) = -2csc2(x)cot(x) |
| csc 2 x | ► Derivative of csc 2 x = -2csc2(x)cot(x) |
| (cscx)^2 | ► Derivative of (cscx)^2 = -2csc2(x)cot(x) |
| csc squared x | ► Derivative of csc squared x = -2csc2(x)cot(x) |
| cscx2 | ► Derivative of cscx2 = -2csc2(x)cot(x) |
| csc^2 | ► Derivative of csc^2 = -2csc2(x)cot(x) |
The Second Derivative Of csc^2x
To calculate the second derivative of a function, differentiate the first derivative.
From above, we found that the first derivative of csc^2x = -2csc2(x)cot(x). So to find the second derivative of csc^2x, we need to differentiate -2csc2(x)cot(x).
We can use the product and chain rules, and then simplify to find the derivative of -2csc2(x)cot(x) is 4csc2(x)cot2(x) + 2csc4(x)
► The second derivative of csc^2x is 4csc2(x)cot2(x) + 2csc4(x)
Interesting property of the derivative of csc^2x
It is interesting to note that the derivative of csc2x is equal to the derivative of cot2x.
The derivative of:
> csc2x = -2.csc2(x).cot(x)
> cot2x = -2.csc2(x).cot(x)
