Derivatives of Inverse Trigonometric and Logarithmic Functions

Derivatives of Inverse Trigonometric Functions

  • arcsin(x)

    • Derivative: ddx[arcsin⁡(x)]=11−x2\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: ddx[arccos⁡(x)]=−11−x2\frac{d}{dx}[\arccos(x)] = -\frac{1}{\sqrt{1 - x^2}}
    • Domain: -1 < x < 1
    • Range: 0 < y < \pi

  • arctan(x)

    • Derivative: ddx[arctan⁡(x)]=11+x2\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∣≥1|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∣≥1|x| \geq 1
    • Range: -\frac{\pi}{2} < y < 0, \ 0 < y < \frac{\pi}{2}

Derivatives of Logarithmic Functions

  • Natural Logarithm (ln)

    • Derivative: ddx[ln⁡(x)]=1x\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: ddx[log⁡a(x)]=1xln⁡(a)\frac{d}{dx}[\log_a(x)] = \frac{1}{x \ln(a)}
    • Conditions: where a > 0 and a<br/>eq1a <br /> eq 1
    • Conversion: Can be related to natural logarithm: log⁡a(x)=ln⁡(x)ln⁡(a)\log_a(x) = \frac{\ln(x)}{\ln(a)}