Derivative of the Cosecant Function

The derivative of the cosecant function is given by $$ D[\csc \ x] = -\frac{\cos x}{\sin^2 x} = -\csc \ x \cdot \cot \ x. $$

Proof

The cosecant function is defined as the reciprocal of the sine function:

$$ \csc x = \frac{1}{\sin x}. $$

Therefore, we can express the derivative of the cosecant function as:

$$ D[\csc \ x] = D\left[ \frac{1}{\sin x} \right]. $$

Applying the quotient rule, we have:

$$ D[\csc \ x] = \frac{D[1] \cdot \sin x - 1 \cdot D[\sin x]}{\sin^2 x}. $$

Since the derivative of a constant is zero, we have $D[1] = 0$.

And the derivative of the sine function is $\cos x$, so $D[\sin x] = \cos x$.

Substituting these values gives:

$$ D[\csc \ x] = \frac{0 \cdot \sin x - 1 \cdot \cos x}{\sin^2 x}. $$

Which simplifies to:

$$ D[\csc \ x] = -\frac{\cos x}{\sin^2 x}. $$

We can rewrite this expression as follows:

$$ D[\csc \ x] = -\frac{\cos x}{\sin x} \cdot \frac{1}{\sin x}. $$

In trigonometry, the ratio $\cos x / \sin x$ is the cotangent function:

$$ D[\csc \ x] = -\cot x \cdot \frac{1}{\sin x}. $$

Since the reciprocal of sine is cosecant, we have:

$$ D[\csc \ x] = -\cot x \cdot \csc x. $$

Thus, we have derived the formula for the derivative of the cosecant function.

And so on.