Derivative of csc(2x) with Proofs and Graphs

The derivative of a cosecant function of double angles results in a composite function. The derivative of the cosecant of 2x is equal to minus two cosecant of 2x times cotangent of 2x, -2csc(2x)cot(2x). This derivative can be found by using the chain rule and the derivative of cosecant of x.

In this article, we’ll look at how to find the composite function cosecant of double angles. We will go over some fundamentals, definitions, formulas, graph comparisons of underived and derived csc(2x), proofs, derivation techniques, and some examples.

CALCULUS
Derivative of cosecant of 2x, csc(2x)

Relevant for

Learning how to find the derivative cosecant of 2x.

See proof

CALCULUS
Derivative of cosecant of 2x, csc(2x)

Relevant for

Learning how to find the derivative cosecant of 2x.

See proof

Proof of the derivative of the cosecant of double angles using chain rule

Because the cosecant of a double angle is a composite function, the chain rule formula is utilized as the basis to derive it. In the composite function csc(2x), the trigonometric function cosecant will be the outer function f(u), while the monomial 2x will be the inner function g(x).

As a prerequisite for this topic, please review the chain rule formula by looking at this article: Chain Rule of derivatives. You may also check out this article for the proof of the derivative of the cosecant function: Derivative of Cosecant, csc(x).

Let’s have the derivative of the function

$latex F(x) = \csc{(2x)}$

We can determine the two functions that comprise F(x). In this instance, there is a trigonometric function cosecant and a monomial. The above function shows that cosecant function is clearly the outer function, whereas the monomial 2x is the inner function. We may configure the outer function as follows:

$latex f(u) = \csc{(u)}$

where

$latex u = 2x$

Setting the monomial 2x as the inner function of f(u) by denoting it as g(x), we have

$latex f(u) = f(g(x))$

$latex g(x) = 2x$

$latex u = g(x)$

Deriving the outer function f(u) using the derivative of cosecant in terms of u, we have

$latex f(u) = \csc{(u)}$

$latex f'(u) = -\csc{(u)}\cot{(u)}$

Deriving the inner function g(x) using power rule since it is a monomial, we have

$latex g(x) = 2x$

$latex g'(x) = 2$

Algebraically multiplying the derivative of outer function $latex f'(u)$ by the derivative of inner function $latex g'(x)$, we have

$latex \frac{dy}{dx} = f'(u) \cdot g'(x)$

$latex \frac{dy}{dx} = (-\csc{(u)}\cot{(u)}) \cdot (2)$

Substituting u into f'(u), we have

$latex \frac{dy}{dx} = (-\csc{(u)}\cot{(u)}) \cdot (2)$

$latex \frac{dy}{dx} = (-\csc{(2x)}\cot{(2x)}) \cdot (2)$

$latex \frac{dy}{dx} = -(\csc{(2x)}\cot{(2x)}) \cdot (2)$

$latex \frac{dy}{dx} = – (2) \cdot (\csc{(2x)}\cot{(2x)})$

In this scenario, we choose not to use the double angle trigonometric identity for cosecant since it complicates the derivative formula. As a result, we arrive at the csc(2x) derivative formula.

$latex \frac{d}{dx} \csc{(2x)} = -2\csc{(2x)}\cot{(2x)}$


How to derive the Cosecant of a Double Angle?

As previously stated, the cosecant of a double angle is a combination of the trigonometric function cosecant and the monomial 2x. This function is straightforward to calculate, and instead of using the chain rule method all the time, we may just apply the proven derivative formula for the cosecant of a double angle.

METHOD 1: When the cosecant of a double angle 2x is to be derived in terms of the same variable x.

$latex \frac{d}{dx} \left( \csc{(2x)} \right) = -2\csc{(2x)}\cot{(2x)}$

Step 1: Analyze if $latex \csc{(2x)}$ is a function of the same variable $latex x$ or f(x). If $latex \csc{(2x)}$ is a function of other variables such as f(t) or f(y), it will use implicit differentiation which is out of the scope of this article.

Step 2: Directly apply the proven derivative formula of the cosecant of a double angle.

$latex \frac{dy}{dx} = -2\csc{(2x)}\cot{(2x)}$

If nothing is to be simplified anymore, that would be the final answer.

METHOD 2: When the given is a cosecant of a function $latex v \times 2$ and to be derived in terms of x.

$latex \frac{d}{dx} \left( \csc{(2v)} \right) = -2\csc{(2v)}\cot{(2v)} \cdot \frac{d}{dx} (v)$

Step 1: Express the function as $latex G(x) = \csc{(2v)}$, where $latex v$ represents any function other than x.

Step 2: Consider $latex \csc{(2x)}$ as the outside function $latex g(v)$ and $latex v$ as the inner function $latex h(x)$ of the composite function $latex G(x)$. Hence we have

$latex g(v) = \csc{(2v)}$

and

$latex h(x) = v$

Step 3: Get the derivative of the outer function $latex g(v)$, which must use the derivative of the cosecant of a double angle, in terms of $latex v$.

$latex \frac{d}{du} \left( \csc{(2v)} \right) = -2\csc{(2v)}\cot{(2v)}$

Step 4: Get the derivative of the inner function $latex h(x) = v$. Use the appropriate derivative rule that applies to $latex v$.

Step 5: Apply the basic chain rule formula by algebraically multiplying the derivative of outer function $latex g(v)$ by the derivative of inner function $latex h(x)$

$latex \frac{dy}{dx} = \frac{d}{du} (g(v)) \cdot \frac{d}{dx} (h(x))$

$latex \frac{dy}{dx} = (-2\csc{(2v)}\cot{(2v)}) \cdot \frac{d}{dx} (v)$

Step 6: Substitute $latex v$ into $latex g'(v)$

Step 7: Simplify and apply any function law whenever applicable to finalize the answer.


Graph of csc(2x) VS. its derivative

Having the function

$latex f(x) = \csc{(2x)}$

its graph illustrates

graph-of-fx-csc2x

And as we know by now, deriving $latex f(x) = \csc{(2x)}$ will lead to

$latex f'(x) = -2\csc{(2x)}\cot{(2x)}$

which if graphically illustrated is

graph-of-derivative of-csc2x

Illustrating both graphs in one, we have

graph-of-csc2x and its derivative

Based on the differences between these graphs, you can conclude that the original function $latex f(x) = \csc{(2x)}$ has a domain of

$$\left(-2\pi,-\frac{3\pi}{2}\right) \cup \left(-\frac{3\pi}{2},-\pi\right) \cup \left(-\pi,-\frac{\pi}{2}\right) \cup \left(-\frac{\pi}{2},0\right)$$

$$\cup \left(0,\frac{\pi}{2}\right) \cup \left(\frac{\pi}{2},\pi\right) \cup \left(\pi,\frac{3\pi}{2}\right) \cup \left(\frac{3\pi}{2},2\pi\right)$$

within the finite intervals of

$latex \left(-2\pi,2\pi\right)$

and exists within the range of

$latex (-\infty,-1] \cup [1,\infty)$

whereas the derivative $latex f'(x) = -2\csc{(2x)}\cot{(2x)}$ has a domain of

$$\left(-2\pi,-\frac{3\pi}{2}\right) \cup \left(-\frac{3\pi}{2},-\pi\right) \cup \left(-\pi,-\frac{\pi}{2}\right) \cup \left(-\frac{\pi}{2},0\right)$$

$$\cup \left(0,\frac{\pi}{2}\right) \cup \left(\frac{\pi}{2},\pi\right) \cup \left(\pi,\frac{3\pi}{2}\right) \cup \left(\frac{3\pi}{2},2\pi\right)$$

within the finite intervals of

$latex \left(-2\pi,2\pi\right)$

and exists within the range of

$latex (-\infty,\infty)$

Graph Comparison between csc(2x) and csc(x) as well as their derivatives

The graphs below demonstrate the difference between $latex \csc{(2x)}$ and $latex \csc{(x)}$.

graph-of-csc2x-vs-cscx

and in terms of their derivatives

graph-of-the-derivatives-of-csc2x-vs-cscx

Examples

The following are some examples of calculating the cosecant of a double angle using either the first or second method, depending on which is most appropriate.

EXAMPLE 1

Derive: $latex f(\beta) = \csc{(2\beta)}$

Solution:

Step 1: Analyzing the given cosecant of a double angle, it is to be derived in terms of $latex \beta$. Therefore, we can use the first method to derive this problem.

Step 2: Directly apply the derivative formula of the cosecant of a double angle in terms of $latex \beta$. Since no further simplification is needed, the final answer is:

$latex f'(\beta) = -2\csc{(2\beta)}\cot{(2\beta)}$

EXAMPLE 2

Derive: $latex G(x) = \csc{(2e^x})$

Solution: Analyzing the given cosecant of a function times two, it is a cosecant of an exponential function multiplied by two. Therefore, we can use the second method to derive this problem.

Step 1: Express the function as $latex G(x) = \csc{(2v)}$, where $latex v$ represents any function other than x. In this problem,

$latex v = e^x$

We will substitute this later as we finalize the derivative of the problem.

Step 2: Consider $latex \csc{(2v)}$ as the outside function $latex g(v)$ and $latex v$ as the inner function $latex h(x)$ of the composite function $latex G(x)$. For this problem, we have

$latex g(v) = \csc{(2v)}$

and

$latex h(x) = v = e^x$

Step 3: Get the derivative of the outer function $latex g(v)$, which must use the derivative of the cosecant of a double angle, in terms of $latex v$.

$$\frac{d}{du} \left( \csc{(2v)} \right) = -2\csc{(2v)}\cot{(2v)}$$

Step 4: Get the derivative of the inner function $latex h(x)$ or $latex v$. Since our $latex v$ in this problem is an exponential function, we will use the derivative of exponential functions to derive $latex v$.

$latex \frac{d}{dx}(h(x)) = \frac{d}{dx} (e^x)$

$latex \frac{d}{dx}(h(x)) = e^x$

Step 5: Apply the basic chain rule formula by algebraically multiplying the derivative of outer function $latex g(v)$ by the derivative of inner function $latex h(x)$

$$\frac{dy}{dx} = \frac{d}{du} (g(v)) \cdot \frac{d}{dx} (h(x))$$

$$\frac{dy}{dx} = (-2\csc{(2v)}\cot{(2v)}) \cdot (e^x)$$

Step 6: Substitute $latex v$ into $latex g'(v)$

$$\frac{dy}{dx} = (-2\csc{(2v)}\cot{(2v)}) \cdot (e^x)$$

$$\frac{dy}{dx} = -(2\csc{(2(e^x))}\cot{(2(e^x))}) \cdot (e^x)$$

Step 7: Simplify and apply any function law whenever applicable to finalize the answer. The final answer is:

$latex G'(x) = -2e^x\csc{(2e^x)}\cot{(2e^x)}$


See also

Interested in learning more about the derivatives of trigonometric functions of 2x? Take a look at these pages:

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