8-2 Practice the Reciprocal Function Family Form G Answers

Derivative Rules

The Derivative tells us the slope of a role at whatsoever point.

There are rules we can follow to find many derivatives.

For example:

• The slope of a abiding value (like iii) is always 0
• The slope of a line like 2x is 2, or 3x is 3 etc
• and then on.

Hither are useful rules to help you work out the derivatives of many functions (with examples below). Annotation: the niggling mark ' means derivative of, and f and m are functions.

Common Functions	Office	Derivative
Abiding	c	0
Line	x	1
	ax	a
Square	x2	2x
Square Root	√x	(½)10-½
Exponential	ex	eastwardx
	ax	ln(a) ax
Logarithms	ln(10)	i/ten
	loga(x)	1 / (x ln(a))
Trigonometry (x is in radians)	sin(x)	cos(x)
	cos(x)	−sin(x)
	tan(x)	sec2(x)
Inverse Trigonometry	sin-1(x)	i/√(1−xtwo)
	cos-1(ten)	−i/√(1−102)
	tan-1(x)	1/(i+x2)
Rules	Function	Derivative
Multiplication by constant	cf	cf'
Power Dominion	xnorth	nxn−ane
Sum Dominion	f + one thousand	f' + grand'
Deviation Rule	f - grand	f' − g'
Product Rule	fg	f g' + f' g
Quotient Dominion	f/yard	f' g − g' f k2
Reciprocal Dominion	1/f	−f'/f2
Concatenation Rule (as "Composition of Functions")	f º g	(f' º thou) × yard'
Concatenation Rule (using ' )	f(one thousand(x))	f'(g(10))g'(x)
Concatenation Rule (using d dx )	dy dx = dy du du dx

"The derivative of" is also written d dx

So d dx sin(x) and sin(ten)' both mean "The derivative of sin(x)"

Examples

Example: what is the derivative of sin(x) ?

From the table above information technology is listed as beingness cos(ten)

It tin be written every bit:

d dx sin(x) = cos(x)

Or:

sin(ten)' = cos(x)

Ability Rule

Example: What is d dx x³ ?

The question is request "what is the derivative of x³ ?"

We can utilize the Power Rule, where n=iii:

d dx x^{due north} = nxⁿ⁻¹

d dx x³ = 3x³⁻¹ = 3xⁱⁱ

(In other words the derivative of x³ is 3x²)

And so it is but this:

"multiply by power
and then reduce power by 1"

It tin as well be used in cases like this:

Example: What is d dx (1/x) ?

1/x is also ten^-one

We can employ the Power Rule, where n = −1:

d dx xⁿ = nx^{northward−1}

d dx ten^-i = −1x^-1−1

= −x^-2

= −i x²

And so we just did this:

which simplifies to −i/x²

Multiplication by constant

Example: What is d dx 5x³ ?

the derivative of cf = cf'

the derivative of 5f = 5f'

We know (from the Power Rule):

d dx x³ = 3x³⁻¹ = 3x²

So:

d dx 5x³ = 5 d dx x³ = 5 × 3x^two = 15x^two

Sum Rule

Instance: What is the derivative of x²+x³ ?

The Sum Rule says:

the derivative of f + g = f' + thou'

So we tin can work out each derivative separately and then add together them.

Using the Power Rule:

• d dx ten² = 2x
• d dx 10^three = 3xⁱⁱ

Then:

the derivative of ten^two + x³ = 2x + 3x²

Difference Rule

What we differentiate with respect to doesn't accept to exist ten, it could be annihilation. In this instance v:

Example: What is d dv (five^three−v^four) ?

The Departure Rule says

the derivative of f − g = f' − g'

Then we can work out each derivative separately and then decrease them.

Using the Ability Rule:

• d dv five³ = 3vⁱⁱ
• d dv v⁴ = 4v^three

And then:

the derivative of v^three − 5⁴ = 3v² − 4v³

Sum, Difference, Constant Multiplication And Ability Rules

Example: What is d dz (5z² + z³ − 7z^four) ?

Using the Power Rule:

• d dz z² = 2z
• d dz zⁱⁱⁱ = 3z²
• d dz z^four = 4z³

Then:

d dz (5z² + z³ − 7z^four) = five × 2z + 3z² − 7 × 4z³
= 10z + 3z² − 28z³

Product Rule

Instance: What is the derivative of cos(x)sin(x) ?

The Production Rule says:

the derivative of fg = f g' + f' g

In our case:

• f = cos
• thousand = sin

We know (from the table above):

• d dx cos(10) = −sin(ten)
• d dx sin(x) = cos(x)

And then:

the derivative of cos(x)sin(ten) = cos(x)cos(ten) − sin(10)sin(x)

= cos²(x) − sin²(x)

Quotient Rule

To assist you lot remember:

( f g )' = gf' − fg' g²

The derivative of "High over Low" is:

"Low dHigh minus High dLow, over the line and square the Low"

Instance: What is the derivative of cos(x)/10 ?

In our instance:

• f = cos
• g = x

Nosotros know (from the table above):

• f' = −sin(x)
• yard' = ane

So:

the derivative of cos(x) x = Low dHigh minus High dLow square the Depression

= ten(−sin(ten)) − cos(x)(ane) x^two

= − xsin(10) + cos(ten) x^two

Reciprocal Dominion

Example: What is d dx (1/x) ?

The Reciprocal Rule says:

the derivative of 1 f = −f' f²

With f(10)= x, nosotros know that f'(ten) = i

So:

the derivative of 1 x = −1 x²

Which is the same result we got above using the Power Dominion.

Chain Rule

Example: What is d dx sin(ten²) ?

sin(10ⁱⁱ) is made up of sin() and x^two :

• f(thou) = sin(g)
• chiliad(x) = x²

The Chain Rule says:

the derivative of f(g(x)) = f'(g(ten))g'(x)

The individual derivatives are:

• f'(g) = cos(g)
• g'(x) = 2x

So:

d dx sin(x²) = cos(g(x)) (2x)

= 2x cos(x²)

Another way of writing the Chain Rule is: dy dx = dy du du dx

Allow's do the previous case again using that formula:

Case: What is d dx sin(ten^two) ?

dy dx = dy du du dx

Let u = x², and then y = sin(u):

d dx sin(x²) = d du sin(u) d dx x²

Differentiate each:

d dx sin(x²) = cos(u) (2x)

Substitute back u = x² and simplify:

d dx sin(x²) = 2x cos(x²)

Same outcome as before (thank goodness!)

Another couple of examples of the Chain Rule:

Case: What is d dx (ane/cos(x)) ?

1/cos(x) is made up of 1/g and cos():

• f(g) = one/g
• one thousand(x) = cos(x)

The Chain Rule says:

the derivative of f(chiliad(ten)) = f'(g(x))one thousand'(ten)

The private derivatives are:

• f'(g) = −i/(g²)
• g'(x) = −sin(10)

So:

(1/cos(x))' = −1 g(ten)² (−sin(x))

= sin(x) cosⁱⁱ(x)

Notation: sin(ten) cos²(x) is likewise tan(x) cos(ten) or many other forms.

Example: What is d dx (5x−2)^three ?

The Chain Rule says:

the derivative of f(g(x)) = f'(one thousand(10))g'(x)

(5x−ii)³ is fabricated up of gⁱⁱⁱ and 5x−two:

• f(g) = g³
• g(x) = 5x−2

The individual derivatives are:

• f'(g) = 3gⁱⁱ (past the Power Rule)
• g'(10) = 5

And so:

d dx (5x−2)^three = (3g(x)ⁱⁱ)(5) = 15(5x−2)ⁱⁱ

6800, 6801, 6802, 6803, 6804, 6805, 6806, 6807, 6808, 6809, 6810, 6811, 6812

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Source: https://www.mathsisfun.com/calculus/derivatives-rules.html

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