How to Find the Derivative of csc(x) Using the Quotient Rule and Chain Rule in Mathematics

d/dx cscx

To find the derivative of csc(x) with respect to x, we can use the quotient rule and the chain rule

To find the derivative of csc(x) with respect to x, we can use the quotient rule and the chain rule.

The function csc(x) can be expressed as 1/sin(x). So, we have:

csc(x) = 1/sin(x)

To find the derivative, let’s use the quotient rule:

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

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

In our case, g(x) is 1 and h(x) is sin(x). Therefore, g'(x) is 0 (since the derivative of a constant is zero) and h'(x) is cos(x) (as the derivative of sin(x) is cos(x)).

Now, applying the quotient rule, we have:

csc'(x) = (0 * sin(x) – 1 * cos(x)) / [sin(x)]^2

csc'(x) = – cos(x) / [sin(x)]^2

Simplifying, we can use the trigonometric identity: 1 + cot(x)^2 = csc(x)^2, where cot(x) = cos(x)/sin(x).

So, we can rewrite the expression as:

csc'(x) = – cos(x) / [sin(x)]^2 = -1 / (sin(x) * [sin(x)]^2) = -1 / sin(x) / sin(x)

csc'(x) = -1 / sin^2(x)

Therefore, the derivative of csc(x) is -1/sin^2(x).

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