Differentiation of Trigonometric Functions

Basic Trigonometric Derivatives:

  1. Derivative of Sine Function:

    • ddx[sin(x)]=cos(x)
  2. Derivative of Cosine Function:

    • ddx[cos(x)]=sin(x)
  3. Derivative of Tangent Function:

    • ddx[tan(x)]=sec2(x)
  4. Derivative of Cosecant Function:

    • ddx[csc(x)]=csc(x)cot(x)
  5. Derivative of Secant Function:

    • ddx[sec(x)]=sec(x)tan(x)
  6. Derivative of Cotangent Function:

    • ddx[cot(x)]=csc2(x)

Using Trigonometric Identities:

  1. Reciprocal Identities:

    • csc(x)=1sin(x)
    • sec(x)=1cos(x)
    • cot(x)=1tan(x)
  2. Pythagorean Identities:

    • sin2(x)+cos2(x)=1
    • tan2(x)+1=sec2(x)
    • 1+cot2(x)=csc2(x)

Derivatives of Inverse Trigonometric Functions:

  1. Derivative of Arcsine Function:

    • ddx[arcsin(x)]=11x2
  2. Derivative of Arccosine Function:

    • ddx[arccos(x)]=11x2
  3. Derivative of Arctangent Function:

    • ddx[arctan(x)]=11+x2
  4. Derivative of Arcsecant Function:

    • ddx[arcsec(x)]=1xx21

Chain Rule with Trigonometric Functions:

When differentiating compositions involving trigonometric functions, the chain rule is applied. For instance:

  • ddx[sin(3x2)]=cos(3x2)ddx[3x2]=3xcos(3x2)