# Stewart – Calculus – 3.5 – Implicit Differentiation and Derivatives of Inverse Trigonometric Functions

Find $y''$ by implicit differentiation:
$9x^{2}+y^{2}=9$.

$18x+2yy'=0$
$2yy'=-18x$
$y'=-9x/y$

$y''=-9(\frac{y\cdot 1\text{--}x\cdot y'}{y^{2}})$

$y''=-9(\frac{y\cdot 1\text{--}x(-9x/y)}{y^{2}})$ ($y'$ is replaced with $-9x/y$)

$y''=-9(\frac{y^{2}+9x^{2}}{y^{3}})$ (after multiplying by $y$ to eliminate denominator from $-9x$)

$y''=-9(\frac{9}{y^{3}})$ (notice that $y^{2}+9x^{2}$ equals the original equation)

So, $y''=\frac{-81}{y^{3}}$.

Derivatives of Inverse Trigonometric Functions

Derivatives of Inverse Trigonometric Functions

# Stewart – Calculus – 3.4 – The Chain Rule

Find the derivative: $y=xe^{-\mathit{kx}}$.

# Stewart – Calculus – 3.3 – Derivatives of Trigonometric Functions

$\frac{d}{\mathit{dx}}(\sin x)=?$ $\frac{d}{\mathit{dx}}(\cos x)=?$ $\frac{d}{\mathit{dx}}(\tan x)=?$ $\frac{d}{\mathit{dx}}(\mathrm{csc} x)=?$ $\frac{d}{\mathit{dx}}(\mathrm{sec} x)=?$ $\frac{d}{\mathit{dx}}(\mathrm{cot} x)=?$
THE TRICK

To learn the derivatives of the 6 trig functions  you actually only have to learn 3 of them. The three to learn are sine, tangent, and secant.

 IF THEN F(X) = sin X F'(X) = cos X F(X) = tan X F'(X) = sec2 X F(X) = sec X F'(X) = sec X tan X

Cosine, Cotangent, and Cosecant.
Think of the functions as having partners:

 sine to cosine tangent to cotangent secant to cosecant

To find the derivative of the "co" functions, start with the derivatives of sine, tangent and secant, change each function in the derivative to it’s co-function partner and put a minus sign in front of the derivative.
So starting with one of the three we need to know:

If F(x) = secant X
then F'(x) = secant X tangent X

So now to get the derivative of the cosecant:
If F(X) = cosecant X
then F'(X) = -cosecant X cotangent X

So all 6 trig function derivatives look like this:

 IF: THEN: F(X) = sinX F'(X) = cosX F(X) = cosX F'(X) = – sinX F(X) = tanX F'(X) = sec2 X F(X) = cotX F'(X) = – csc2 X F(X) = secX F'(X) = sec X tan X F(X) = cscX F'(X) = -csc X cot X

Thanks to Bruce Kirkpatrick for this.

# Stewart – Calculus – 3.2 – Differentiation Rules

THE PRODUCT RULE

$\frac{d}{\mathit{dx}}[f(x)g(x)]=f(x)\frac{d}{\mathit{dx}}[g(x)]+g(x)\frac{d}{\mathit{dx}}[f(x)]$

(a) If $f(x)=xe^{x}$, find $f'(x)$.

(b) Find the nth derivative, $f^{(n)}(x)$.

Differentiate the function $f(t)=\sqrt{t}(a+\mathit{bt})$

If $f(x)=\sqrt{x} \hspace{1 mm} g(x)$, where $g(4)=2$ and $g'(4)=3$, find $f'(4)$.

THE QUOTIENT RULE

$\frac{d}{\mathit{dx}}[\frac{f(x)}{g(x)}]=\frac{g(x)\frac{d}{\mathit{dx}}[f(x)]\text{--}f(x)\frac{d}{\mathit{dx}}[g(x)]}{[g(x)]^{2}}$

Let $y=\frac{x^{2}+x\text{--}2}{x^{3}+6}$

Find an equation of the tangent line to the curve $y=\frac{e^{x}}{(1+x^{2})}$ at the point $(1,\frac{1}{2}e)$.

$\frac{d}{\mathit{dx}}(c)=?$ $\frac{d}{\mathit{dx}}(x^{n})=?$ $\frac{d}{\mathit{dx}}(e^{x})=?$ $(\mathit{cf})'=?$ $(f+g)'=?$ $(f-g)'=?$ $(\mathit{fg})'=?$ $(\frac{f}{g})'=?$
SOLUTION 1
Using the Product Rule

SOLUTION 2
If we first use the laws of exponents to rewrite f(t), then we can proceed directly without using the Product Rule.

0

# Stewart – Calculus – 3.1 – Derivatives of Polynomials and Exponential Functions

(a) $f(x)=\frac{1}{x^{2}}$

(b) $y=\sqrt[{3}]{x^{2}}$

Find equations of the tangent line and normal line to the curve $y=x\sqrt{x}$ at the point $(1, 1)$. Illustrate by graphing the curve and these lines.

THE CONSTANT MULTIPLE RULE

$\frac{d}{\mathit{dx}}(3x^{4})$ $\frac{d}{\mathit{dx}}(-x)$

# Stewart – Calculus – 2.7 – Derivatives and Rates of Change

1. Find an equation of the tangent line to the parabola $y=x^{2}$ at the point $P(1,1)$.
2. Find an equation of the tangent line to the hyperbola $f(x)=\frac{3}{x}$ at the point $(3, 1)$.
3. Find the derivative of the function $f(x)=x^{2}-8x+9$ at the number $a$.

# Stewart – Calculus – 2.5 – Continuity

1. Definition: A function f is continuous at a number a if

This definition implicitly requires three things if f is continuous at a:

1. f(a) is defined (that is, a is in the domain of f)

2. $\lim_{x \to a}f(x)$ exists

3. $\lim_{x \to a}f(x)=f(a)$

2. If $f$ and $g$ are continuous functions with $f(3)=5$ and
$\lim_{x \to 3}[2f(x) - g(x)]=4$, find $g(3)$.

Use the definition of continuity and the properties of limits to show that the funtion is continous at the given number a. $f(x)=(x+2x^{3})^{4}$, $a=-1$

3. The Intermediate Value Theorem
Suppose that $f$ is continuous on the closed interval $[a, b]$ and let $N$ be any number between $f(a)$ and $f(b)$, where $f(a) \neq f(b)$. Then there exists a number $c$ in $(a,b)$ such that $f(c)=N$.

~ ~ ~

Use the Intermediate Value Theorem to show that there is
a root of the given equation in the specified interval.

$x^4+x-3=0, (1,2)$

# Stewart – Calculus – 2.4 – The Precise Definition of a Limit

1. Provide the symbolic equivalent.

a. every

b. there exist(s)

c. such that

2. Write the precise definition of a limit.

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.

# Calculus Test 1

1. Specify the domain of the function y = |x – 1|.

2. Which of the following is a true statement about the graph of the equation $y = x^4 +1$?
3. 1. It is symmetric about the x-axis.
2. It is symmetric about the y-axis.
3. It has two x-intercepts.
4. It has no x-intercepts.

a. Only statements 2 and 3 are true.
b. Only statements 2 and 4 are true.
c. Only statements 1 and 3 are true.
d. Only statement 2 is true.

4. Which of the following statements are true of the graph of $y = \frac{2x-1}{x+1}$?

1. It has no x-intercept.
2. It has a slant asymptote y=2x.
3. It has a vertical asymptote at $x=-1$.
4. It has a horizontal asymptote at y=2.

a. Statements 1, 2, and 3 are true.
b. Statements 3 and 4 are true.
c. Statements 2, 3, and 4 are true.
d. All four statements are true.

5. Let $f(x) = 2 \cos x$. The domain of $f^{-1}(x)$ is

1. $[-1,1]$ 2. $(2, \infty)$ 3. $(-\infty, \infty)$ 4. $(-2,2)$ 5. $[-2,2]$

6. Simplify as far as possible. $\ln e +a^{\log_{a}5}- \log100+10^0 - \log_{3} \frac{1}{3}$
7. Solve for x.

a. $\log_3 x + \log_3(2x+5)=1$

b. $\frac{1}{\sqrt{2}}=4^{x}$

8. $f(x)=(x-3)(x+1)^2(x-1)^4$
9. Identify the parts of the following composite function. $f(g(h(j(x))))=\frac{1}{\sqrt{\log(x-1)}}$
10. Consider the picture below. Find the equation of the line L.

11. The following table represents a function of the form $f(x)=ab^{x}$. Find the equation of the function.

$\begin{tabular}{| l | c | r |} \hline x & f(x)\\ \hline 0 & 6\\ \hline 1 & 18\\ \hline 2 & 54\\ \hline 3 & 162\\ \hline 4 & 486\\ \hline \end{tabular}$
12. Given $a^{m}=2$, $a^{n}=3$, $b^{m}=4$, and $b^{n}=5$,
use the properties of exponentials to determine $(a^{3n}b^{m+n})^{\frac{1}{3}}$.

13. Evaluate:
1. $\sin \frac{\pi}{3}$
2. $\tan \frac{3\pi}{4}$
3. $\cos \frac{5\pi}{6}$
14. Find all solutions to the equaion, $\tan^{2}x=\tan x$, such that $x \in [0,2\pi]$.
15. Find $f^{-1}(x)$ if $f(x) = \sqrt{e^{x}+2}$.