2023-11-06
Find derivatives of continuous functions with one variable
Find antiderivatives and integrals of functions with one variable
Example 1.1
\(f(x) = 2\)
Derivative of a constant
\[\dfrac{d}{dx} c = 0\]
Example 1.2
\(f(x) = 2x\)
Example 1.3
\(f(x) = 2x+2\)
Example 1.4
\(f(x) = x^2\)
Derivative of \(x\) to a constant
\[\dfrac{d}{dx} x^n = nx^{n-1}\]
Example 1.5
\(f(x) = 3\sqrt{x}+\frac2x+5\)
Example 1.6
\(f(x) = e^x\)
Derivative of exponential function
\(\dfrac{d}{dx} e^x = e^x\)
Example 1.7
\(f(x) = \ln(x)\)
Derivative of logarithm
\(\dfrac{d}{dx} ln(x) = \dfrac{1}{x}\)
Example 1.8
\(f(x) = x^2 e^x\)
Product Rule
\(\dfrac{d}{dx} f(x)g(x) = f'(x)g(x) + f(x)g'(x)\)
Example 1.9
\(f(x) = \frac{x^5}{2x+7}\)
Quotient Rule
\(\dfrac{d}{dx} \dfrac{f(x)}{g(x)} = \dfrac{g(x)f'(x) - f(x)g'(x)}{\big(g(x)\big)^2}\)
Example 1.10
\(f(x) = e^{-2x+7}\)
Chain Rule
\(\dfrac{d}{dx} f\big(g(x)\big)= f'\big(g(x)\big)g'(x)\)
Example 1.11
\(f(x) = \ln(x^2)\)
Example 2.1
\(f(x) = 2\)
Example 2.2
\(f(x) = x\)
Integration of x to a constant
\(\displaystyle\int x^n dx = \dfrac{x^{n+1}}{n+1} + c\)
Example 2.3
\(f(x) = \frac1x\)
Example 2.4
\(f(x) = x^{3/2}\)
Example 2.5
\(f(x) = e^x\)
Example 2.6
\(f(x) = e^{-x}\)
Example 2.7
\(f(x) = e^{-2x}\)
Example 3.1
\(\displaystyle\int_0^1 (2x+x^5)dx\)
Example 3.2
\(\displaystyle\int_2^3 e^{-x}dx\)
U-substitution
\(\displaystyle\int f\big(g(x)\big) g'(x) dx = \displaystyle\int f(u) dx\)
Example 3.3
\(\displaystyle\int_2^3 x e^{x^2}dx\)
Example 3.4
\(\displaystyle\int_0^{\infty} x e^{-x}dx\)
Integrating by Parts
\(\displaystyle\int f(x) g'(x) dx = f(x)g(x) - \\ \displaystyle\int f'(x) g(x) dx\)
OR
\(\displaystyle\int_a^b u dv = uv\bigg|^b_a - \displaystyle\int_a^b vdu\)
Example 3.5
\(\displaystyle\int_1^2 x^2 \ln(x)dx\)
Example 3.6
\(\displaystyle\int_1^2 \ln(x)dx\)
Example 3.7
\(\displaystyle\int_1^2 x^2 e^{x}dx\)
Chapter 24 Slides