Derivatives of Inverse Trigonometric and Logarithmic Functions

Derivatives of Inverse Trigonometric Functions

• arcsin(x)

  • Derivative: \frac{d}{dx}[\arcsin(x)] = \frac{1}{\sqrt{1 - x^2}}
  • Domain: -1 < x < 1
  • Range: -\frac{\pi}{2} < y < \frac{\pi}{2}

• arccos(x)

  • Derivative: \frac{d}{dx}[\arccos(x)] = -\frac{1}{\sqrt{1 - x^2}}
  • Domain: -1 < x < 1
  • Range: 0 < y < \pi

• arctan(x)

  • Derivative: \frac{d}{dx}[\arctan(x)] = \frac{1}{1 + x^2}
  • Domain: -\infty < x < \infty
  • Range: -\frac{\pi}{2} < y < \frac{\pi}{2}

• arccot(x)

  • Derivative: \frac{d}{dx}[\arccot(x)] = -\frac{1}{1 + x^2}
  • Domain: -\infty < x < \infty
  • Range: 0 < y < \pi

• arcsec(x)

  • Derivative: \frac{d}{dx}[\arcsec(x)] = \frac{1}{|x|\sqrt{x^2 - 1}}
  • Domain: |x| \geq 1
  • Range: 0 < y < \frac{\pi}{2}, \frac{\pi}{2} < y < \frac{3\pi}{2}

• arccsc(x)

  • Derivative: \frac{d}{dx}[\arccsc(x)] = -\frac{1}{|x|\sqrt{x^2 - 1}}
  • Domain: |x| \geq 1
  • Range: -\frac{\pi}{2} < y < 0, \ 0 < y < \frac{\pi}{2}

Derivatives of Logarithmic Functions

• Natural Logarithm (ln)

  • Derivative: \frac{d}{dx}[\ln(x)] = \frac{1}{x}
  • Domain: x > 0
  • Application: Useful for solving problems involving exponential growth and decay.

• Logarithm to any base (log_a(x))

  • Derivative: \frac{d}{dx}[\log_a(x)] = \frac{1}{x \ln(a)}
  • Conditions: where a > 0 and $$a
    eq 1$$
  • Conversion: Can be related to natural logarithm: \log_a(x) = \frac{\ln(x)}{\ln(a)}