How to Find the Derivative of csc(x) using the Chain Rule: Step-by-Step Guide with Trigonometric Identities

d/dx(csc(x))

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

To find the derivative of csc(x), we can use the chain rule. The derivative of csc(x) with respect to x is given by:

d/dx(csc(x)) = -csc(x)cot(x)

Let’s break down the steps to derive this result:

Step 1: Start with the function csc(x).

Step 2: Rewrite csc(x) as 1/sin(x). We know that csc(x) is the reciprocal of sin(x).

Step 3: Apply the quotient rule. The quotient rule states that for functions u(x) = f(x)/g(x), the derivative is given by:

d/dx(u(x)) = (g(x)f'(x) – f(x)g'(x))/[g(x)]^2

In this case, f(x) = 1 and g(x) = sin(x). Therefore, we have:

d/dx(csc(x)) = (sin^2(x)(-1) – 1(cos(x)))/[sin(x)]^2

Step 4: Simplify the expression.

Since sin^2(x) = 1 – cos^2(x), we can rewrite the numerator as:

-(1 – cos^2(x) – cos(x))

= -1 + cos^2(x) + cos(x)

Step 5: Substitute back into the quotient rule expression:

d/dx(csc(x)) = -1 + cos^2(x) + cos(x) / [sin(x)]^2

Step 6: Rewrite cos^2(x) as 1 – sin^2(x):

d/dx(csc(x)) = -1 + (1 – sin^2(x)) + cos(x) / [sin(x)]^2

Now, simplify further:

= -1 + 1 – sin^2(x) + cos(x) / [sin(x)]^2

= -sin^2(x) + cos(x) / [sin(x)]^2

Step 7: Recall the trigonometric identity that sin^2(x) + cos^2(x) = 1:

= -1 – cos(x) / [sin(x)]^2

Finally, rewrite the expression:

d/dx(csc(x)) = -csc(x)cot(x)

Therefore, the derivative of csc(x) with respect to x is -csc(x)cot(x).

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