10/8/2021

## 3.4 Derivative Of E^f(x) And Ln (f(x))ap Calculus

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The Derivative tells us the slope of a function at any point.

Compute the derivatives of the following functions (a) f(x) = x 2 + 1 x − 3.03x + 4 (b) g(x) = ln(x) (c) h(x) = x 2 (x 3 + 4)5 (d) f(x) = x2 x (e) f(x) = e 4x (f) f(x) = e x 1 + x 2. Let f(x) = x 3 − 3x 2 + 0.5x − 2. (a) Compute the average rate of change of f(x) from x = −1 to x = 1. (b) Draw an accurate graph of f(x) with x between. Play this game to review Calculus. Find dy/dx Preview this quiz on Quizizz. Derivative of ln x and e^x DRAFT. Find the second derivative of f(x).

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
Constantc0
Linex1
axa
Squarex22x
Square Root√x(½)x
Exponentialexex
axln(a) ax
Logarithmsln(x)1/x
loga(x)1 / (x ln(a))
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−1
Sum Rulef + gf’ + g’
Difference Rulef - gf’ − g’
Product Rulefgf g’ + f’ g
Quotient Rulef/g(f’ g − g’ f )/g2
Reciprocal Rule1/f−f’/f2
Chain 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)'

## Examples

### Example: 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)

### Example: 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)

So it is simply this:

3x^2'>
'multiply by power
then reduce power by 1'

It can also be used in cases like this:

## 3.4 Derivative Of E^f(x) And Ln (f(x))ap Calculus Solver

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/x2

### Example: 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 = 15x2

### Example: 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 + 3x2

### Difference 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 − 4v3

### Example: 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 − 28z3

### Example: 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

(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' = 1

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)x2

### Example: 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.

### Example: 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)

## 3.4 Derivative Of E^f(x) And Ln (f(x))ap Calculus Calculator

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:

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

The Derivative Calculator lets you calculate derivatives of functions online — for free!

Our calculator allows you to check your solutions to calculus exercises. It helps you practice by showing you the full working (step by step differentiation).

The Derivative Calculator supports computing first, second, …, fifth derivatives as well as differentiating functions with many variables (partial derivatives), implicit differentiation and calculating roots/zeros. You can also check your answers! Interactive graphs/plots help visualize and better understand the functions.

For more about how to use the Derivative Calculator, go to 'Help' or take a look at the examples.

And now: Happy differentiating!

Enter the function you want to differentiate into the Derivative Calculator. Skip the 'f(x) =' part! The Derivative Calculator will show you a graphical version of your input while you type. Make sure that it shows exactly what you want. Use parentheses, if necessary, e. g. 'a/(b+c)'.

In 'Examples', you can see which functions are supported by the Derivative Calculator and how to use them.

When you're done entering your function, click 'Go!', and the Derivative Calculator will show the result below.

In 'Options' you can set the differentiation variable and the order (first, second, … derivative). You can also choose whether to show the steps and enable expression simplification.

Clicking an example enters it into the Derivative Calculator. Moving the mouse over it shows the text.

Configure the Derivative Calculator:

The practice problem generator allows you to generate as many random exercises as you want.

You find some configuration options and a proposed problem below. You can accept it (then it's input into the calculator) or generate a new one.

Accept problemNext problem

This will be calculated:

Not what you mean? Use parentheses! Set differentiation variable and order in 'Options'.

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### Result

Above, enter the function to derive. Differentiation variable and more can be changed in 'Options'. Click 'Go!' to start the derivative calculation. The result will be shown further below.

## 3.4 Derivative Of E^f(x) And Ln (f(x))ap Calculus 14th Edition

### How the Derivative Calculator Works

For those with a technical background, the following section explains how the Derivative Calculator works.

First, a parser analyzes the mathematical function. It transforms it into a form that is better understandable by a computer, namely a tree (see figure below). In doing this, the Derivative Calculator has to respect the order of operations. A specialty in mathematical expressions is that the multiplication sign can be left out sometimes, for example we write '5x' instead of '5*x'. The Derivative Calculator has to detect these cases and insert the multiplication sign.

The parser is implemented in JavaScript, based on the Shunting-yard algorithm, and can run directly in the browser. This allows for quick feedback while typing by transforming the tree into LaTeX code. MathJax takes care of displaying it in the browser.

When the 'Go!' button is clicked, the Derivative Calculator sends the mathematical function and the settings (differentiation variable and order) to the server, where it is analyzed again. This time, the function gets transformed into a form that can be understood by the computer algebra systemMaxima.

Maxima takes care of actually computing the derivative of the mathematical function. Like any computer algebra system, it applies a number of rules to simplify the function and calculate the derivatives according to the commonly known differentiation rules. Maxima's output is transformed to LaTeX again and is then presented to the user.

Displaying the steps of calculation is a bit more involved, because the Derivative Calculator can't completely depend on Maxima for this task. Instead, the derivatives have to be calculated manually step by step. The rules of differentiation (product rule, quotient rule, chain rule, …) have been implemented in JavaScript code. There is also a table of derivative functions for the trigonometric functions and the square root, logarithm and exponential function. In each calculation step, one differentiation operation is carried out or rewritten. For example, constant factors are pulled out of differentiation operations and sums are split up (sum rule). This, and general simplifications, is done by Maxima. For each calculated derivative, the LaTeX representations of the resulting mathematical expressions are tagged in the HTML code so that highlighting is possible.

The 'Check answer' feature has to solve the difficult task of determining whether two mathematical expressions are equivalent. Their difference is computed and simplified as far as possible using Maxima. For example, this involves writing trigonometric/hyperbolic functions in their exponential forms. If it can be shown that the difference simplifies to zero, the task is solved. Otherwise, a probabilistic algorithm is applied that evaluates and compares both functions at randomly chosen places.

The interactive function graphs are computed in the browser and displayed within a canvas element (HTML5). For each function to be graphed, the calculator creates a JavaScript function, which is then evaluated in small steps in order to draw the graph. While graphing, singularities (e. g. poles) are detected and treated specially. The gesture control is implemented using Hammer.js.

If you have any questions or ideas for improvements to the Derivative Calculator, don't hesitate to write me an e-mail.

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