Fall 2025: Derivatives of Logarithmic and Inverse Trigonometric Functions

Welcome to a focused exploration of derivatives, specifically targeting logarithmic and inverse trigonometric functions. This section covers essential techniques for differentiating complex function compositions, including combinations of logarithmic, algebraic, and trigonometric functions. Let's build a solid understanding together!

Topics Covered

  • Derivatives of Logarithmic Functions: Understanding the fundamental rules for differentiating logarithmic functions. Recall that the derivative of the natural logarithm function, $\ln(x)$, is given by: $$\frac{d}{dx} \ln(x) = \frac{1}{x}$$ For a general logarithmic function with base $b$, the derivative is: $$\frac{d}{dx} \log_b(x) = \frac{1}{x \ln(b)}$$
  • Derivatives of Combinations: Mastering the chain rule and product/quotient rules in conjunction with logarithmic functions. For example, finding the derivative of $f(x) = \ln(x^6 + \sqrt{7-x})$ requires using the chain rule. If $y = \ln(u)$, then $\frac{dy}{dx} = \frac{u'}{u}$, where $u = x^6 + \sqrt{7-x}$.
  • Logarithmic Differentiation: This powerful technique simplifies differentiation of complex functions involving products, quotients, and exponents. The general approach is to take the natural logarithm of both sides of an equation and then differentiate implicitly. For example, to differentiate $y = \frac{x^{3/4}\sqrt{x^2+1}}{(3x+2)^5}$, take the natural log of both sides: $\ln y = \frac{3}{4}\ln x + \frac{1}{2}\ln(x^2+1) - 5\ln(3x+2)$. Then, differentiate both sides with respect to $x$.
  • Derivatives of Inverse Trigonometric Functions: Memorizing and applying the derivatives of the six basic inverse trigonometric functions. Here are a few key derivatives:
    • $\frac{d}{dx} \arcsin(x) = \frac{1}{\sqrt{1-x^2}}$
    • $\frac{d}{dx} \arccos(x) = -\frac{1}{\sqrt{1-x^2}}$
    • $\frac{d}{dx} \arctan(x) = \frac{1}{1+x^2}$
  • Compositions with Inverse Trigonometric Functions: Applying the chain rule to find derivatives of composite functions involving inverse trigonometric and algebraic functions. For instance, if $y = \arccos(x^4)$, then $y' = -\frac{4x^3}{\sqrt{1-x^8}}$.

Remember, practice is key to mastering these concepts! Work through various examples and don't hesitate to ask questions. You've got this!