1. What is the formula for the derivative of Sine? What about for Cosine? What r

Question

1. What is the formula for the derivative of Sine? What about for Cosine? What role does the chain rule play in these formulas?

2. What is the formula for the derivative of Tangent? What about for Cotangent?

3.  What is the formula for the derivative of Secant? What about for Cosecant?

4. What is the formula for the derivative of ArcSine? What about for ArcCosine?

5. What is the formula for the derivative of ArcTan? What about for ArcCot?

6. What is the formula for the derivative of ArcSec? What about for ArcCsc?

Explanation / Answer

1) derivative of sine x = cosine x
derivative of cosine x = -sine x

Chain Rule: (f(g(x))' = (f(g(x))' = f' (g(x)) . g'(x)

In words: Take the derivative of the outer function, plug in the inner function, and multiply by the

derivative of the inner function.

We make use of chainr rule in finding differentials of the kind sin (3x), cos ( 4x) etc.
Moreover, using chain rule we can find the derivative of cosine using the derivative of sine.

2) derivative of tangent x = (secant^2) x
derivative of Cotangent x = - (cosec ^2) x

3) derivative of Secant x = sec x tan x
  derivative of Cosecant x = - cosec x cot x

4) derivative of arcsin x = 1 / sqrt(1 - x^2)
derivative of ArcCosine x = -1 / sqrt(1 - x^2)

5) derivative of ArcTan x = 1/ (1 + x^2)
   derivative of ArcCot x = -1 / (1 + x^2)

6) derivative of ArcSec x = 1/ ( |x| * sqrt(x^2 - 1) )
derivative of ArcCsc x = -1/ ( |x| * sqrt(x^2 - 1) ).

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