LOGARITHMIC LIMITS AND DERIVATIVES

📋 Course Outline

1. Derivative of Product
2. Primitive of Logarithm
3. Limit of Logarithm at Zero
4. Limit of Logarithm at Infinity
5. Limit of Functions at Zero and Infinity
6. Limit of Logarithm with Function Limit
7. Limit Calculation Techniques
8. Logarithm Limit Examples

📖 1. Derivative of Product

🔑 Key Concepts & Definitions

• Product Rule: If $u(x)$ and $v(x)$ are differentiable functions, then the derivative of their product is: $(uv)' = u'v + uv'$ This rule allows calculating the derivative of a product by differentiating each function separately and combining the results.

• Primitive (Antiderivative): A function $F$ is a primitive of $f$ if $F' = f$.
  In the context of the product rule, the primitive of $\frac{u'}{u}$ (with $u \neq 0$) is $\ln|u|$.

• Logarithmic Derivative: For a differentiable, non-zero function $u$, the derivative of $\ln|u|$ is: $\frac{d}{dx} \ln|u| = \frac{u'}{u}$ This links derivatives of functions to their logarithms, useful in integration and limit calculations.

• Limit Behavior of Logarithms:

  • $\lim_{x \to 0^+} \ln x = -\infty$
  • $\lim_{x \to +\infty} \ln x = +\infty$
    Understanding these helps evaluate limits involving logarithms.