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1.4: Derivatives of Sums, Products and Quotients
https://math.libretexts.org/Bookshelves/Calculus/Elementary_Calculus_2e_(Corral)/01%3A_The_Derivative/1.04%3A_Derivatives_of_Sums_Products_and_Quotients
\[\begin{aligned} d(f \cdot g) ~&=~ (f \cdot g)(x + \dx) ~-~ (f \cdot g)(x)\\ &=~ f(x + \dx) \cdot g(x + \dx) ~-~ f(x) \cdot g(x) &\\ &=~ \text{(area of outer rectangle)} ~-~ \text{(area of original r...\[\begin{aligned} d(f \cdot g) ~&=~ (f \cdot g)(x + \dx) ~-~ (f \cdot g)(x)\\ &=~ f(x + \dx) \cdot g(x + \dx) ~-~ f(x) \cdot g(x) &\\ &=~ \text{(area of outer rectangle)} ~-~ \text{(area of original rectangle)}\\ &=~ \text{sum of the areas of the three shaded inner rectangles}\\ &=~ f(x) \cdot \dg ~+~ g(x) \cdot \df ~+~ \df \cdot \dg\\ &=~ f(x) \cdot \dg ~+~ g(x) \cdot \df ~+~ (f'(x)\;\dx) \cdot (g'(x)\;\dx)\\ &=~ f(x) \cdot \dg ~+~ g(x) \cdot \df ~+~ (f'(x) g'(x)) \cdot (\dx)^2\\ &=~ f(x) \cdo…
8: Applications of Integrals
https://math.libretexts.org/Bookshelves/Calculus/Elementary_Calculus_2e_(Corral)/08%3A_Applications_of_Integrals
6.1: Integration by Parts
https://math.libretexts.org/Bookshelves/Calculus/Elementary_Calculus_2e_(Corral)/06%3A_Methods_of_Integration/6.01%3A_Integration_by_Parts
\[\begin{aligned} d(x\,e^{-x}) ~&=~ x\,d(e^{-x}) ~+~ d(x)\,e^{-x}\\ &=~ -x\,e^{-x}\,\dx ~+~ e^{-x}\,\dx\\ d(x\,e^{-x}) ~&=~ -x\,e^{-x}\,\dx ~-~ d(e^{-x})\\ x\,e^{-x}\,\dx ~&=~ -d(x\,e^{-x}) ~-~ d(e^{-...\[\begin{aligned} d(x\,e^{-x}) ~&=~ x\,d(e^{-x}) ~+~ d(x)\,e^{-x}\\ &=~ -x\,e^{-x}\,\dx ~+~ e^{-x}\,\dx\\ d(x\,e^{-x}) ~&=~ -x\,e^{-x}\,\dx ~-~ d(e^{-x})\\ x\,e^{-x}\,\dx ~&=~ -d(x\,e^{-x}) ~-~ d(e^{-x}),\quad\text{so integrate both sides to get}\
1.6: Higher Order Derivatives
https://math.libretexts.org/Bookshelves/Calculus/Elementary_Calculus_2e_(Corral)/01%3A_The_Derivative/1.06%3A_Higher_Order_Derivatives
That is, think of $\ddx$ as the differentiation operator on the collection of differentiable functions, taking a function $f(x)$ to its derivative function $\dfdx$ : Likewise, \(\frac{d^2}{d\!x^2...That is, think of $\ddx$ as the differentiation operator on the collection of differentiable functions, taking a function $f(x)$ to its derivative function $\dfdx$ : Likewise, $\frac{d^2}{d\!x^2}$ is an operator on twice-differentiable functions, taking a function $f(x)$ to its second derivative function $\frac{d^2 \negmedspace f}{d\!x^2}$ : In general, an eigenfunction of an operator $A$ is a function $\phi(x)$ such that $A(\phi(x)) ~=~ \lambda \cdot \phi(x)$ , that is, for all …
4.2: Trigonometric Equations
https://math.libretexts.org/Courses/City_University_of_New_York/College_Algebra_and_Trigonometry-_Expressions_Equations_and_Graphs/04%3A_Introduction_to_Trigonometry_and_Transcendental_Expressions/4.02%3A_Trigonometric_Equations
So far, we were concerned only with finding a single solution (say, between 0◦ and 90◦0◦ and 90◦0^◦\text{ and }90^◦ ). In this section we will be concerned with finding the most general solution to s...So far, we were concerned only with finding a single solution (say, between 0◦ and 90◦0◦ and 90◦0^◦\text{ and }90^◦ ). In this section we will be concerned with finding the most general solution to such trigonometric equations with one variable. In particular, we will solve equations with a variable angle.
6: Methods of Integration
https://math.libretexts.org/Bookshelves/Calculus/Elementary_Calculus_2e_(Corral)/06%3A_Methods_of_Integration
2.4: General Exponential and Logarithmic Functions
https://math.libretexts.org/Bookshelves/Calculus/Elementary_Calculus_2e_(Corral)/02%3A_Derivatives_of_Common_Functions/2.04%3A_General_Exponential_and_Logarithmic_Functions
\[\begin{aligned} \ln\,y ~&=~ \ln\,\left(a^x\right) ~=~ x \,\ln\,a\\ \ddx\,(\ln\,y) ~&=~ \ddx\,(x \,\ln\,a) ~=~ \ln\,a\\ \frac{y'}{y} ~&=~ \ln\,a \quad\Rightarrow\quad y' ~=~ y \cdot \ln\,a\end{aligne... $\begin{aligned} \ln\,y ~&=~ \ln\,\left(a^x\right) ~=~ x \,\ln\,a\\ \ddx\,(\ln\,y) ~&=~ \ddx\,(x \,\ln\,a) ~=~ \ln\,a\\ \frac{y'}{y} ~&=~ \ln\,a \quad\Rightarrow\quad y' ~=~ y \cdot \ln\,a\end{aligned} \nonumber$ Thus, the derivative of $y = a^x$ is: In general, for an exponent of the form $u = u(x)$ :
3.2: Limits- Formal Definition
https://math.libretexts.org/Bookshelves/Calculus/Elementary_Calculus_2e_(Corral)/03%3A_Topics_in_Differential_Calculus/3.02%3A_Limits-_Formal_Definition
$-\frac{1}{x} ~\le~ \frac{\sin\,x}{x} ~\le~ \frac{1}{x} ~~\text{for all $x > 0$} \quad\Rightarrow\quad \lim_{x \to \infty}~\frac{\sin\,x}{x} ~=~ 0 \nonumber$ by the Squeeze Theorem, since \(\displa... $-\frac{1}{x} ~\le~ \frac{\sin\,x}{x} ~\le~ \frac{1}{x} ~~\text{for all $x > 0$} \quad\Rightarrow\quad \lim_{x \to \infty}~\frac{\sin\,x}{x} ~=~ 0 \nonumber$ by the Squeeze Theorem, since $\displaystyle\lim_{x \to \infty}~\frac{-1}{x} ~=~ 0 ~=~ \displaystyle\lim_{x \to \infty}~\frac{1}{x}$ .
5.4: Integration by Substitution
https://math.libretexts.org/Bookshelves/Calculus/Elementary_Calculus_2e_(Corral)/05%3A_The_Integral/5.04%3A_Integration_by_Substitution
$\int\,\sec\,x~\dx ~=~ \int\,\frac{\sec\,x~(\sec\,x ~+~ \tan\,x)}{\sec\,x ~+~ \tan\,x}~\dx ~=~ \int\,\frac{\sec\,x\;\tan\,x ~+~ \sec^2 x}{\sec\,x ~+~ \tan\,x}~\dx \nonumber$ and that the numerator ... $\int\,\sec\,x~\dx ~=~ \int\,\frac{\sec\,x~(\sec\,x ~+~ \tan\,x)}{\sec\,x ~+~ \tan\,x}~\dx ~=~ \int\,\frac{\sec\,x\;\tan\,x ~+~ \sec^2 x}{\sec\,x ~+~ \tan\,x}~\dx \nonumber$ and that the numerator in the last integral is the derivative of the denominator: let $u = \sec\,x ~+~ \tan\,x$ , so that $\du = (\sec\,x\;\tan\,x ~+~ \sec^2 x)\,\dx$ .
7.4: Translations and Rotations
https://math.libretexts.org/Bookshelves/Calculus/Elementary_Calculus_2e_(Corral)/07%3A_Analytic_Geometry_and_Plane_Curves/7.04%3A_Translations_and_Rotations
Conversely, to transform a second-degree equation in $x$ and $y$ into a "standard" conic section equation in terms of $x^{\prime}$ and $y^{\prime}$ (to simplify sketching the graph), the "reve...Conversely, to transform a second-degree equation in $x$ and $y$ into a "standard" conic section equation in terms of $x^{\prime}$ and $y^{\prime}$ (to simplify sketching the graph), the "reverse" rotation equations for $x$ and $y$ in terms of $x^{\prime}$ and $y^{\prime}$ are needed:
6.6: Numerical Integration Methods
https://math.libretexts.org/Bookshelves/Calculus/Elementary_Calculus_2e_(Corral)/06%3A_Methods_of_Integration/6.06%3A_Numerical_Integration_Methods
In the above command, the statement x(3:2:end-1) allows you to skip every other element in the list x after position 3, by moving up the list in increments of 2 positions all the way to the next-to-la...In the above command, the statement x(3:2:end-1) allows you to skip every other element in the list x after position 3, by moving up the list in increments of 2 positions all the way to the next-to-last position in the list (end-1). Note that Simpson’s rule gives essentially the true value in this case, and the value from the trapezoid rule is virtually the same as the value produced by the built-in trapz function in Octave/MATLAB:

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