Polynomial Functions, Limits, and Determinants in Calculus I

Apuntes

## Mathematical Functions and Derivatives • Subtopic: Polynomial Functions and Derivatives The function $f(x) = (3x^3)^2$ has a derivative $f'(x) = 6x^6$. Another expression, $(3x^3 + 6)^{1/2}$, simplifies to $f(x) = 54x^2$. For the function $f(x) = a^u$, the derivative is $f'(x) = a^u \ln a \cdot n^{n-1} \cdot u'$, where $u$ is a polynomial. For the function $f(x) = 3(x-6)(x+2)$, the derivative is $f'(x) = 12(x^2-6x+2)(2x-6)$. The expression $2^x f'(x) = \frac{1}{2}(3x^3-6)^{-1/2} 9x^2$ is another example of differentiation. The expression $a^{n^m} = a^{n \cdot m}$ is a rule for exponents. The derivative $f'(x) = 9$ is a constant function. The expression $4 \sqrt{x^2-6}$ simplifies to $\frac{4}{4}(x^2-6)^{1/2}$. The derivative $f'(x) = \frac{q}{8}(x^2-6)^{-3/2} \cdot 4x^3$ is another example. The expressions $1/a^2 = a^{-2}$ and $1/a^n = a^{-n}$ are rules for negative exponents. • Subtopic: Determinants and Matrices The determinant of a second-order matrix is given by: $$ \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \\ \end{vmatrix} $$ The determinant $D$ is calculated as $D = a_1 \cdot b_2 - a_2 \cdot b_1$. For example, the determinant $\Delta$ of the matrix: $$ \begin{vmatrix} 5 & -3 ...