Question

Below is an attempt to derive the derivative of csc x using the product rule, where x is in the domain of csc x. In which step, if any, does an error first appear?

Step 1: csc x sin x = 1
Step 2: d/dx(csc x -sin x)= 0
Step 3: d/dx(csc x) - sin x + csc x - cos x = 0
Step 4: d/dx (cscx)=
(-csc x - cos x)/sin x = -csc x - cot x

a) Step 1
b) Step 2
c) Step 3
d) There is no error

Answer 1

The error first appears in step c where the product rule is incorrectly applied.

Step-by-step explanation:

The error first appears in step c. In step 3, the application of the product rule is incorrect. The correct application of the product rule would be:

1. d/dx(csc(x)⋅sin(x)) = (d/dx(csc(x)))⋅sin(x) + csc(x)⋅(d/dx(sin(x)))

After applying the correct product rule, you can then simplify the expression to obtain the derivative of csc(x).

Therefore, the correct derivative of csc(x) using the product rule is -csc(x)⋅cot(x).