The derivatives of the sine, cosine, and tangent functions are really important when studying calculus. Let’s break this down into simpler parts!

1. Sine Function:

• The derivative of the sine function is:
  • If you take the derivative of sin x, you get cos x.

2. Cosine Function:

• The derivative of the cosine function is:
  • If you take the derivative of cos x, you get -sin x.

3. Tangent Function:

• The derivative of the tangent function is:
  • If you take the derivative of tan x, you get sec^2 x.

Key Comparisons:

• Sine and Cosine:

  • The derivative of sine (which is cos x) and the derivative of cosine (which is -sin x) both repeat their values in a regular pattern between 0 and 2π.

• Tangent:

  • The derivative of tangent (which is sec^2 x) grows really fast. This can happen when sec x becomes very big at certain points.

• Overall Behavior:

  • The derivatives for sine and cosine show a smooth, wavy motion.
  • On the other hand, the derivative for tangent shows a sudden increase in value near specific points called asymptotes.

And that’s it! We can see how these functions change and behave through their derivatives.