Calculus 1 Chapter 2 Flashcards

Questions and Answers

What is the derivative of $y=5x^2$?

5 times derivative of $x^2$

What is the derivative of a regular number?

ZERO

What is the derivative of $(x^2 + 3)^6$ using the chain rule?

2x times 6 $(x^2+3)^5$

What is the derivative product rule?

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What is the derivative of $x^{-6}$?

-6x^{-7} Signup and view all the answers

What is the derivative of a quotient $(x^2-3)^3 / (x^{10}-10x^2)^5$?

=$(x^2-3)^3 imes (x^{10}-10x^2)^{-5}$ Signup and view all the answers

What is the derivative of $e^x$?

= $e^x$ Signup and view all the answers

What is the derivative of $ ext{ln}(x)$?

1/x Signup and view all the answers

What is the derivative of $x^{-1}$?

ln x Signup and view all the answers

What is the derivative of $ ext{sin}(x)$?

cos(x) Signup and view all the answers

What is the derivative of $ ext{cos}(x)$?

-sin(x) Signup and view all the answers

What is the derivative of $ ext{tan}(x)$?

1/cos(x)^2 Signup and view all the answers

What is $x^{-1}$?

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How is $ ext{cos}^5(x)$ expressed?

($ ext{cos}(x)$)^5 Signup and view all the answers

What is the formula to find the slope-intercept form of a derivative?

y-y1=m(x-x1) Signup and view all the answers

What is the alternative formula for differentiation of $|x-1|$?

f'(c) = ext{lim} rac{f(x)-f(c)}{x-c}$ as $x$ approaches c Signup and view all the answers

What is $5/(2x)^3$ simplified?

$5/8 imes x'^{3}$ Signup and view all the answers

What is the derivative of $7/(3x)^{-2}$?

126x Signup and view all the answers

What is the quotient rule?

f'(x)g(x) - g'(x)f(x) / (g(x))^2 Signup and view all the answers

What do you get when you rewrite $rac{1/x}{x+5}$ by multiplying everything by x?

x/(x^2 + 5x) Signup and view all the answers

What is the derivative of $8(x^2 + 6)$?

derivative of $8x^2 + 48$ which is the same as $8x^2$ Signup and view all the answers

How do you find the differential of anything?

First, find the derivative which is $f'$, then calculate it out of $f' = dy/dx$; thus, $dy = dx imes f'$ Signup and view all the answers

What does approximating function values mean?

Finding $ ext{√26}$ or $23^2$ or any formula Signup and view all the answers

Study Notes

Derivatives Overview

Derivative of ( y = 5x^2 ) is calculated as ( 5 \times \frac{d}{dx}(x^2) ).
The derivative of a constant (regular number) is always zero.

Chain Rule

For the function ( (x^2 + 3)^6 ), the derivative is computed using:
- ( 2x \times 6(x^2 + 3)^5 ).

Product and Quotient Rules

Product Rule:
- The derivative of a product of two functions ( f(x) ) and ( g(x) ) is ( f'(x)g(x) + f(x)g'(x) ).
Quotient Rule applies to functions like ( \frac{(x^2-3)^3}{(x^{10}-10x^2)^5} ) and is represented as:
- ( \frac{u'v - uv'}{v^2} ), where ( u ) and ( v ) are the numerator and denominator functions, respectively.

Common Derivatives

Derivative of ( e^x ) is ( e^x ).
Derivative of ( \ln x ) is ( \frac{1}{x} ).
Derivative of ( x^{-1} ) is ( -1/x^2 ).
Derivative of ( \sin x ) is ( \cos x ).
Derivative of ( \cos x ) is ( -\sin x ).
Derivative of ( \tan x ) is ( \frac{1}{\cos^2 x} ) (also known as ( \sec^2 x )).

Function Transformations

( x^{-1} ) can be equated to ( \frac{1}{x} ).
( \cos^5 x ) can be expressed as ( (\cos x)^5 ).

Slope-Intercept Form

The formula for finding the slope-intercept form is ( y - y_1 = m(x - x_1) ).

Limits and Differentiation

For differentiation of absolute values like ( |x - 1| ):
- ( f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c} ).

Simplifying Expressions

Expression ( \frac{5}{(2x)^3} ) simplifies to ( \frac{5}{8} x^{-3} ).
For ( \frac{7}{(3x)^{-2}} ), the derivative evaluates to ( 126x ).

Rewriting Complex Functions

A complex function like ( \frac{1}{x}/(x + 5) ) can be rewritten by multiplying through by ( x ):
- Results in ( \frac{x}{x^2 + 5x} ), facilitating easier differentiation.

Differential Concepts

Find differentials by calculating derivatives: ( f' = \frac{dy}{dx} ).
Actual differential ( dy ) is expressed as ( dy = dx \times f' ).

Approximating Function Values

Approximating functions involves finding values like ( \sqrt{26} ) or ( 23^2 ) within the context of their formulas.
For instance, ( \sqrt{26} ) can be approximated using ( \sqrt{25} + ) differential.

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Test your understanding of derivatives with these flashcards from Calculus 1 Chapter 2. Each card includes a term related to differentiation and its definition, helping you reinforce your calculus knowledge. Perfect for quick revision and study sessions.