Derivative of Cosecant Function (csc(x)) using Quotient Rule in Calculus

d/dx csc x

To find the derivative of csc(x), we can use the quotient rule

To find the derivative of csc(x), we can use the quotient rule.

The quotient rule states that if we have a function f(x) = g(x)/h(x), then its derivative is given by:

f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2

In this case, g(x) = 1 and h(x) = sin(x). Therefore, g'(x) = 0 and h'(x) = cos(x).

Applying the quotient rule, we get:

d/dx(csc(x)) = (0 * sin(x) – 1 * cos(x)) / (sin(x))^2

Simplifying the numerator:

d/dx(csc(x)) = -cos(x) / (sin(x))^2

The final result is:

d/dx(csc(x)) = -cos(x) / sin^2(x) or -cot(x)csc(x)

Now, let’s define some terms related to calculus:

1. Derivative: The derivative of a function measures how the function changes as its input (usually denoted by x) changes. It represents the rate of change of the function at a particular point. It is denoted by f'(x) or dy/dx.

2. Quotient rule: The quotient rule is a rule in calculus that allows us to find the derivative of a function that can be expressed as the quotient of two functions. It is given by (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2.

3. Trigonometric functions: Trigonometric functions are mathematical functions that relate the angles of a triangle to the ratios of the lengths of its sides. They include functions like sin(x), cos(x), tan(x), csc(x), sec(x), and cot(x).

4. Cosecant function (csc(x)): The cosecant function is the reciprocal of the sine function. It is defined as csc(x) = 1/sin(x), where sin(x) ≠ 0. It gives the ratio of the hypotenuse to the side opposite an angle in a right triangle.

5. Cotangent function (cot(x)): The cotangent function is the reciprocal of the tangent function. It is defined as cot(x) = 1/tan(x), where tan(x) ≠ 0. It gives the ratio of the adjacent side to the opposite side an angle in a right triangle.

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