Compute the differential \(dy\) for \(y = x^2 + 3x − 5\)
Solve using the power rule, the sum rule and the difference rule.
\[ \begin{align} dy &= f'(x^2 + 3x − 5)dx \\ &= (2x + 3)dx \end{align} \]
library(Deriv)
fun1 <- function(x){
x^2 + 3*x - 5
}
fun1prime <- Deriv(fun1)
fun1prime## function (x)
## 2 * x + 3Compute the differential \(dy\) for \(y = x^7 − x^5\)
Solve using the power rule and the difference rule
\[ \begin{align} dy &= f'(x^7 − x^5)dx \\ &= (7x^6 - 5x^4)dx \end{align} \]
library(Deriv)
fun2 <- function(x){
(x^7) - (x^5)
}
fun2prime <- Deriv(fun2)
fun2prime## function (x)
## {
## .e1 <- x^2
## x^4 * (2 * .e1 + 5 * (.e1 - 1))
## }Compute the differential \(dy\) for \(y = \frac{1}{4x^2}\)
Solve using the reciprocal rule.
\[ \begin{align} dy &= f'(\frac{1}{4x^2})dx \\ &= \frac{-8x}{16x^4} dx\\ &= -\frac{1}{2x^3} dx \end{align} \]
library(Deriv)
fun3 <- function(x){
1 / (4*x^2)
}
fun3prime <- Deriv(fun3)
fun3prime## function (x)
## -(8 * (x/(4 * x^2)^2))Compute the differential \(dy\) for \(y = (2x + \sin x)^2\)
Solve using the chain rule and trig rules.
\[ \begin{align} dy &= f'((2x + \sin x)^2)dx \\ &= (2(2x + \sin x) \cdot (2 + \cos x))dx \\ &= (8x + 4\sin x + 4x\cos x + 2\sin x \cos x) dx\\ &= (8x + 4\sin x + 4x\cos x + \sin(2x)) dx \end{align} \]
library(Deriv)
fun4 <- function(x){
(2*x + sin(x))^2
}
fun4prime <- Deriv(fun4)
fun4prime## function (x)
## 2 * ((2 * x + sin(x)) * (2 + cos(x)))Compute the differential \(dy\) for \(y = x^2e^{3x}\)
Solve using the product rule.
\[ \begin{align} dy &= f'(x^2e^{3x})dx \\ &= (2x \space e^{3x} + x^2 \space 3e^{3x}) dx \\ &= ((2x + 3x^2)e^{3x}) dx \end{align} \]
library(Deriv)
fun5 <- function(x){
x^2 * exp(1)^(3*x)
}
fun5prime <- Deriv(fun5)
fun5prime## function (x)
## {
## .e1 <- 2.71828182845905^(3 * x)
## x * (2 * .e1 + 3 * (.e1 * x))
## }Compute the differential \(dy\) for \(y= \frac{4}{x^4}\)
Solve using the quotient rule.
\[ \begin{align} dy &= f'(\frac{4}{x^4})dx \\ &= (\frac{0 \cdot x^4 - 4 \cdot 4x^3}{{x^4}^2}) dx \\ &= (\frac{-16x^3}{x^8}) dx \\ &= (-\frac{16}{x^5}) dx \end{align} \]
library(Deriv)
fun6 <- function(x){
4/x^4
}
fun6prime <- Deriv(fun6)
fun6prime## function (x)
## -(16/x^5)