Download here: http://gg.gg/virku
The Derivative tells us the slope of a function at any point.
There are rules we can follow to find many derivatives.
For example:
*The slope of a constant value (like 3) is always 0
*The slope of a line like 2x is 2, or 3x is 3 etc
*and so on.
Here are useful rules to help you work out the derivatives of many functions (with examples below). Note: the little mark ’ means ’Derivative of’, and f and g are functions.Common FunctionsFunction
Derivative
Constantc0Linex1axaSquarex22xSquare Root√x(½)x-½Exponentialexexaxln(a) axLogarithmsln(x)1/xloga(x)1 / (x ln(a))Trigonometry (x is in radians)sin(x)cos(x)cos(x)−sin(x)tan(x)sec2(x)Inverse Trigonometrysin-1(x)1/√(1−x2)cos-1(x)−1/√(1−x2)tan-1(x)1/(1+x2)RulesFunction
Derivative
Multiplication by constantcfcf’Power Rulexnnxn−1Sum Rulef + gf’ + g’Difference Rulef - gf’ − g’Product Rulefgf g’ + f’ gQuotient Rulef/g(f’ g − g’ f )/g2Reciprocal Rule1/f−f’/f2Chain Rule
(as ’Composition of Functions’)f º g(f’ º g) × g’Chain Rule (using ’ )f(g(x))f’(g(x))g’(x)Chain Rule (using ddx )dydx = dydududx
’The derivative of’ is also written ddx
So ddxsin(x) and sin(x)’ both mean ’The derivative of sin(x)’
Search this site. Many of the rules for calculating derivatives of real-valued functions can be applied to calculating the derivatives of vector-valued functions as well. Recall that the derivative of a real-valued function can be interpreted as the slope of a tangent line or the instantaneous rate of change of the function.ExamplesExample: what is the derivative of sin(x) ?
From the table above it is listed as being cos(x)
It can be written as:
sin(x) = cos(x)
Or:
sin(x)’ = cos(x)Power RuleExample: What is x3 ?
The question is asking ’what is the derivative of x3 ?’
We can use the Power Rule, where n=3:
xn = nxn−1
x3 = 3x3−1 = 3x2
(In other words the derivative of x3 is 3x2)
Two different penguin groups who are existed on ice fields started a sweet war with each other. Their aims are dropping their opponents into the cold sea water by shooting with some different weapons. After game has loaded assign game’s graphic quality from ’ Detail Level ’ section. You can use ’ HIGH ’ selection for this. Penguin wars ps4.
So it is simply this: Unit 3 Derivative Rules Of Compositesap Calculus Integrals
3x^2’>
’multiply by power
then reduce power by 1’
It can also be used in cases like this:Example: What is (1/x) ?
1/x is also x-1
We can use the Power Rule, where n = −1:
xn = nxn−1
x−1 = −1x−1−1
= −x−2
= −1x2
So we just did this:
-x^-2’>
which simplifies to −1/x2Multiplication by constantExample: What is 5x3 ?
the derivative of cf = cf’
the derivative of 5f = 5f’
We know (from the Power Rule):
x3 = 3x3−1 = 3x2
So:
5x3 = 5x3 = 5 × 3x2 = 15x2Sum RuleExample: What is the derivative of x2+x3 ?
The Sum Rule says:
the derivative of f + g = f’ + g’
So we can work out each derivative separately and then add them.
Using the Power Rule:
*x2 = 2x
*x3 = 3x2
And so:
the derivative of x2 + x3 = 2x + 3x2Difference Rule
It doesn’t have to be x, we can differentiate with respect to, for example, v: Example: What is (v3−v4) ?
The Difference Rule says
the derivative of f − g = f’ − g’
So we can work out each derivative separately and then subtract them.
Using the Power Rule:
*v3 = 3v2
*v4 = 4v3
And so:
the derivative of v3 − v4 = 3v2 − 4v3Sum, Difference, Constant Multiplication And Power RulesExample: What is (5z2 + z3 − 7z4) ?
Using the Power Rule:
*z2 = 2z
*z3 = 3z2
*z4 = 4z3
And so:
(5z2 + z3 − 7z4) = 5 × 2z + 3z2 − 7 × 4z3 = 10z + 3z2 − 28z3Product RuleExample: What is the derivative of cos(x)sin(x) ?
The Product Rule says:
the derivative of fg = f g’ + f’ g
In our case:
*f = cos
*g = sin
We know (from the table above):
*cos(x) = −sin(x)
*sin(x) = cos(x)
So:
the derivative of cos(x)sin(x) = cos(x)cos(x) − sin(x)sin(x)
= cos2(x) − sin2(x)Quotient Rule
To help you remember:
(fg)’ = gf’ − fg’g2
The derivative of ’High over Low’ is:
’Low dHigh minus High dLow, over the line and square the Low’
Example: What is the derivative of cos(x)/x ?
In our case:
*f = cos
*g = x
We know (from the table above):
*f’ = −sin(x)
*g’ = 1Unit 3 Derivative Rules Of Compositesap Calculus 2nd Edition
So:
the derivative of cos(x)x = Low dHigh minus High dLowover the line and square the Low
= x(−sin(x)) − cos(x)(1)x2
= −xsin(x) + cos(x)x2Reciprocal RuleExample: What is (1/x) ?
The Reciprocal Rule says:
the derivative of 1f = −f’f2
With f(x)= x, we know that f’(x) = 1
So:
the derivative of 1x = −1x2
Which is the same result we got above using the Power Rule.Chain RuleExample: What is ddxsin(x2) ?
sin(x2) is made up of sin() and x2:
*f(g) = sin(g)
*g(x) = x2
The Chain Rule says:
the derivative of f(g(x)) = f’(g(x))g’(x)
The individual derivatives are:
*f’(g) = cos(g)
*g’(x) = 2x
So:
ddxsin(x2) = cos(g(x)) (2x)
= 2x cos(x2)
Another way of writing the Chain Rule is: dydx = dydududx
Let’s do the previous example again using that formula:Example: What is ddxsin(x2) ?
dydx = dydududx
Have u = x2, so y = sin(u):
ddx sin(x2) = ddusin(u)ddxx2
Differentiate each:
ddx sin(x2) = cos(u) (2x)
Substitue back u = x2 and simplify:
ddx sin(x2) = 2x cos(x2)
Same result as before (thank goodness!)
Another couple of examples of the Chain Rule:Example: What is (1/cos(x)) ?
1/cos(x) is made up of 1/g and cos():
*f(g) = 1/g
*g(x) = cos(x)
The Chain Rule says:
the derivative of f(g(x)) = f’(g(x))g’(x)
The individual derivatives are:
*f’(g) = −1/(g2)
*g’(x) = −sin(x)
So:
(1/cos(x))’ = −1/(g(x))2 × −sin(x)
= sin(x)/cos2(x)
Note: sin(x)/cos2(x) is also tan(x)/cos(x), or many other forms.Example: What is (5x−2)3 ?
The Chain Rule says:Unit 3 Derivative Rules Of Compositesap Calculus Solver
the derivative of f(g(x)) = f’(g(x))g’(x)
(5x-2)3 is made up of g3 and 5x-2:
*f(g) = g3
*g(x) = 5x−2
The individual derivatives are:
*f’(g) = 3g2 (by the Power Rule)
*g’(x) = 5
So:
(5x−2)3 = 3g(x)2 × 5 = 15(5x−2)2
Unit 3 – Derivatives
_________________________________________________________________
Student Learning Objectives for Unit 3:
Upon completion of Unit 3, students will be able to
*Conceptualize derivative presented graphically, numerically, and analytically.
*Interpret derivative as an instantaneous rate of change
*Calculate slopes and derivatives using the definition of a derivative.
*Graph f from the graph of f’, graph f’ from the graph of f, and graph the derivative of a function given numerically with data.
*Articulate and identify corresponding characteristic of graphs of f and f’
*Determine where a function is not differentiable and distinguish between corners, cusps, discontinuities, and vertical tangents.
*Approximate derivatives from graphs and tables of values
*Know derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.
*Use sum, product, quotient, chain rules to calculate derivative of composite functions.
*Use derivative to calculate the instantaneous rate of change.
*Use derivatives to analyze straight line motion and solve other problems involving rates of change.
*Articulate and identify corresponding characteristic of graphs of f, f’, and f”
*Use the graph of f” to identify the points of inflection and concavity of f.
*Use the graph of f’ to identify the local (relative) extrema and the increasing/decreasing behavior of f.
*Find slopes of parameterized curves
*Find derivatives using implicit differentiation.
*Find derivatives using the Power Rule for Rational Powers of x.
Download here: http://gg.gg/virku
https://diarynote-jp.indered.space
The Derivative tells us the slope of a function at any point.
There are rules we can follow to find many derivatives.
For example:
*The slope of a constant value (like 3) is always 0
*The slope of a line like 2x is 2, or 3x is 3 etc
*and so on.
Here are useful rules to help you work out the derivatives of many functions (with examples below). Note: the little mark ’ means ’Derivative of’, and f and g are functions.Common FunctionsFunction
Derivative
Constantc0Linex1axaSquarex22xSquare Root√x(½)x-½Exponentialexexaxln(a) axLogarithmsln(x)1/xloga(x)1 / (x ln(a))Trigonometry (x is in radians)sin(x)cos(x)cos(x)−sin(x)tan(x)sec2(x)Inverse Trigonometrysin-1(x)1/√(1−x2)cos-1(x)−1/√(1−x2)tan-1(x)1/(1+x2)RulesFunction
Derivative
Multiplication by constantcfcf’Power Rulexnnxn−1Sum Rulef + gf’ + g’Difference Rulef - gf’ − g’Product Rulefgf g’ + f’ gQuotient Rulef/g(f’ g − g’ f )/g2Reciprocal Rule1/f−f’/f2Chain Rule
(as ’Composition of Functions’)f º g(f’ º g) × g’Chain Rule (using ’ )f(g(x))f’(g(x))g’(x)Chain Rule (using ddx )dydx = dydududx
’The derivative of’ is also written ddx
So ddxsin(x) and sin(x)’ both mean ’The derivative of sin(x)’
Search this site. Many of the rules for calculating derivatives of real-valued functions can be applied to calculating the derivatives of vector-valued functions as well. Recall that the derivative of a real-valued function can be interpreted as the slope of a tangent line or the instantaneous rate of change of the function.ExamplesExample: what is the derivative of sin(x) ?
From the table above it is listed as being cos(x)
It can be written as:
sin(x) = cos(x)
Or:
sin(x)’ = cos(x)Power RuleExample: What is x3 ?
The question is asking ’what is the derivative of x3 ?’
We can use the Power Rule, where n=3:
xn = nxn−1
x3 = 3x3−1 = 3x2
(In other words the derivative of x3 is 3x2)
Two different penguin groups who are existed on ice fields started a sweet war with each other. Their aims are dropping their opponents into the cold sea water by shooting with some different weapons. After game has loaded assign game’s graphic quality from ’ Detail Level ’ section. You can use ’ HIGH ’ selection for this. Penguin wars ps4.
So it is simply this: Unit 3 Derivative Rules Of Compositesap Calculus Integrals
3x^2’>
’multiply by power
then reduce power by 1’
It can also be used in cases like this:Example: What is (1/x) ?
1/x is also x-1
We can use the Power Rule, where n = −1:
xn = nxn−1
x−1 = −1x−1−1
= −x−2
= −1x2
So we just did this:
-x^-2’>
which simplifies to −1/x2Multiplication by constantExample: What is 5x3 ?
the derivative of cf = cf’
the derivative of 5f = 5f’
We know (from the Power Rule):
x3 = 3x3−1 = 3x2
So:
5x3 = 5x3 = 5 × 3x2 = 15x2Sum RuleExample: What is the derivative of x2+x3 ?
The Sum Rule says:
the derivative of f + g = f’ + g’
So we can work out each derivative separately and then add them.
Using the Power Rule:
*x2 = 2x
*x3 = 3x2
And so:
the derivative of x2 + x3 = 2x + 3x2Difference Rule
It doesn’t have to be x, we can differentiate with respect to, for example, v: Example: What is (v3−v4) ?
The Difference Rule says
the derivative of f − g = f’ − g’
So we can work out each derivative separately and then subtract them.
Using the Power Rule:
*v3 = 3v2
*v4 = 4v3
And so:
the derivative of v3 − v4 = 3v2 − 4v3Sum, Difference, Constant Multiplication And Power RulesExample: What is (5z2 + z3 − 7z4) ?
Using the Power Rule:
*z2 = 2z
*z3 = 3z2
*z4 = 4z3
And so:
(5z2 + z3 − 7z4) = 5 × 2z + 3z2 − 7 × 4z3 = 10z + 3z2 − 28z3Product RuleExample: What is the derivative of cos(x)sin(x) ?
The Product Rule says:
the derivative of fg = f g’ + f’ g
In our case:
*f = cos
*g = sin
We know (from the table above):
*cos(x) = −sin(x)
*sin(x) = cos(x)
So:
the derivative of cos(x)sin(x) = cos(x)cos(x) − sin(x)sin(x)
= cos2(x) − sin2(x)Quotient Rule
To help you remember:
(fg)’ = gf’ − fg’g2
The derivative of ’High over Low’ is:
’Low dHigh minus High dLow, over the line and square the Low’
Example: What is the derivative of cos(x)/x ?
In our case:
*f = cos
*g = x
We know (from the table above):
*f’ = −sin(x)
*g’ = 1Unit 3 Derivative Rules Of Compositesap Calculus 2nd Edition
So:
the derivative of cos(x)x = Low dHigh minus High dLowover the line and square the Low
= x(−sin(x)) − cos(x)(1)x2
= −xsin(x) + cos(x)x2Reciprocal RuleExample: What is (1/x) ?
The Reciprocal Rule says:
the derivative of 1f = −f’f2
With f(x)= x, we know that f’(x) = 1
So:
the derivative of 1x = −1x2
Which is the same result we got above using the Power Rule.Chain RuleExample: What is ddxsin(x2) ?
sin(x2) is made up of sin() and x2:
*f(g) = sin(g)
*g(x) = x2
The Chain Rule says:
the derivative of f(g(x)) = f’(g(x))g’(x)
The individual derivatives are:
*f’(g) = cos(g)
*g’(x) = 2x
So:
ddxsin(x2) = cos(g(x)) (2x)
= 2x cos(x2)
Another way of writing the Chain Rule is: dydx = dydududx
Let’s do the previous example again using that formula:Example: What is ddxsin(x2) ?
dydx = dydududx
Have u = x2, so y = sin(u):
ddx sin(x2) = ddusin(u)ddxx2
Differentiate each:
ddx sin(x2) = cos(u) (2x)
Substitue back u = x2 and simplify:
ddx sin(x2) = 2x cos(x2)
Same result as before (thank goodness!)
Another couple of examples of the Chain Rule:Example: What is (1/cos(x)) ?
1/cos(x) is made up of 1/g and cos():
*f(g) = 1/g
*g(x) = cos(x)
The Chain Rule says:
the derivative of f(g(x)) = f’(g(x))g’(x)
The individual derivatives are:
*f’(g) = −1/(g2)
*g’(x) = −sin(x)
So:
(1/cos(x))’ = −1/(g(x))2 × −sin(x)
= sin(x)/cos2(x)
Note: sin(x)/cos2(x) is also tan(x)/cos(x), or many other forms.Example: What is (5x−2)3 ?
The Chain Rule says:Unit 3 Derivative Rules Of Compositesap Calculus Solver
the derivative of f(g(x)) = f’(g(x))g’(x)
(5x-2)3 is made up of g3 and 5x-2:
*f(g) = g3
*g(x) = 5x−2
The individual derivatives are:
*f’(g) = 3g2 (by the Power Rule)
*g’(x) = 5
So:
(5x−2)3 = 3g(x)2 × 5 = 15(5x−2)2
Unit 3 – Derivatives
_________________________________________________________________
Student Learning Objectives for Unit 3:
Upon completion of Unit 3, students will be able to
*Conceptualize derivative presented graphically, numerically, and analytically.
*Interpret derivative as an instantaneous rate of change
*Calculate slopes and derivatives using the definition of a derivative.
*Graph f from the graph of f’, graph f’ from the graph of f, and graph the derivative of a function given numerically with data.
*Articulate and identify corresponding characteristic of graphs of f and f’
*Determine where a function is not differentiable and distinguish between corners, cusps, discontinuities, and vertical tangents.
*Approximate derivatives from graphs and tables of values
*Know derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.
*Use sum, product, quotient, chain rules to calculate derivative of composite functions.
*Use derivative to calculate the instantaneous rate of change.
*Use derivatives to analyze straight line motion and solve other problems involving rates of change.
*Articulate and identify corresponding characteristic of graphs of f, f’, and f”
*Use the graph of f” to identify the points of inflection and concavity of f.
*Use the graph of f’ to identify the local (relative) extrema and the increasing/decreasing behavior of f.
*Find slopes of parameterized curves
*Find derivatives using implicit differentiation.
*Find derivatives using the Power Rule for Rational Powers of x.
Download here: http://gg.gg/virku
https://diarynote-jp.indered.space
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