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).
More Answers:
Understanding Revenue | How to Calculate and Differentiate it from Profit in Mathematical TermsDerivative of sin(u) with respect to x using the chain rule
How to Find the Derivative of cos(u) with Respect to x | Chain Rule Explained