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Applied calculus topic summary: techniques of differentiation

Tools: Function Evaluator & Grapher | Excel Grapher | Excel First and Second Derivative Grapher

Subtopics: Derivatives of Powers, Sums, and Constant Multiples | Marginal Analysis | Product & Quotient Rules | Calculation Thought Experiment | Chain Rule | Derivatives of Logarithmic and Exponential Functions | Derivatives of Trigonometric Functions | Implicit Functions & Implicit Differentiation

Derivatives of Powers, Sums, and Constant Multiples

Power Rule
If f(x) = xn, then f'(x) = nxn-1. This is true for any real n. In differential notation, this reads

    d

    dx
    xn
    =nxn - 1

Sum and Constant Multiple Rules
If f'(x) and g'(x) exist, and c is a constant, then

(A) [f(x)   ±   g(x)]' = f'(x)   ±   g'(x)
(B) [cf(x)]' = cf'(x) .
In differential notation, these rules are
  (A)
d

dx
[f(x) ± g(x)] =
d

dx
[f(x)] ±
d

dx
[g(x)]
  (B)
d

dx
[cf(x)] = c
d

dx
[f(x)]

In words:

The derivative of a sum is the sum of the derivatives, and the derivative of a difference is the difference of the derivatives.

The derivative of c times a function is c times the derivative of the function.

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Examples

    d

    dx
    x3.2
    =3.2x2.2
    d

    dx
    4x - 1
    = - 4x - 2
    d

    dx
    3x1.1 + 4x - 2
    = 3.3x0.1 - 8x - 3
    d

    dx
    1

    x
    =
    d

    dx
    x - 1
    = - x - 2
    =
    -
    1

    x2

Want some practice? Try the interactive tutorial or try some exercises.

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Marginal Analysis

If Q(x) represents any quantity such as cost, revenue, profit or loss on the sale of x items, then Q'(x) is called the marginal quantity. Thus, for instance, the marginal cost measures the increase in total cost per item. This is effectively the cost of each additional item.

The marginal cost is distinct from the average cost, which measures the average of the total cost of the first x items. Average cost is given by

C(x) =
C(x)

x
=
Total Cost

Number of Items

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Examples

Suppose the cost of (the first) x items is given by

C(x) = 4x0.2 - 0.1x pounds Sterling.

Then the marginal cost is
C'(x) = 0.8x-0.8 - 0.1 pounds Sterling per unit.
In particular, C'(3) ≈ 0.23 (pounds Sterling per unit) is the approximate cost of the third unit (or the fourth unit).

The average cost of the first three units is

    C(3) =
    C(3)

    3
    4.6829

    3
    1.56 pound Sterling/item

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Product Rule
    d

    dx
    [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

Product Rule In Words:

The derivative of a product is the derivative of the first times the second, plus the first times the derivative of the second.


Quotient Rule

    d

    dx
    f(x)

    g(x)
    =
    f'(x)g(x) - f(x)g'(x)

    g(x)2

Quotient Rule In Words:

The derivative of a quotient is the derivative of the top times the bottom, minus the top times the derivative of the bottom, all over the bottom squared.
Examples


Product Rule

    d

    dx
    [x2(3x- 1)] = 2x(3x- 1) + x2(3)

(The derivatives of f and g are shown in color.)

 


Quotient Rule

    d

    dx
    x3

    x2+1
    =
    3x2(x2+1) - x3.2x

    (x2+1)2

Of course, you should simplify the answers and not leave them like that!

Press here for an on-line tutorial on the product & quotient rules.

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Combining Rules for Differentiation: Calculation Thought Experiment

The calculation thought experiment is a technique to determine whether to treat an algebraic expression as a product, quotient, sum, or difference. Given such an expression, consider the steps you would use in computing its value. If the last operation is multiplication, treat the expression as a product; if the last operation is division, treat the expression as a quotient, and so on.

Using the Calculation Thought Experiment (CTE) to Differentiate a Function
If the CTE says, for instance, that the expression is a sum of two smaller expressions, then apply the rule for sums as a first step. This will leave you having to differentiate simpler expressions, and you can use the CTE on these, and so on...

Examples

1. (3x2- 4)(2x+1) can be computed by first calculating the expressions in parentheses and then multiplying. Since the last step is multiplication, we can treat the expression as a product.

2. (2x- 1)/x can be computed by first calculating the numerator and denominator, and then dividing one by the other. Since the last step is division, we can treat the expression as a quotient.

3. x2 + (4x- 1)(x+2) can be computed by first calculating x2, then calculating the product (4x- 1)(x+2), and finally adding the two answers. Thus, we can treat the expression as a sum.

4. (3x2- 1)5 can be computed by first calculating the expression in parentheses, and then raising the answer to the fifth power. Thus, we can treat the expression as a power.

Using the CTE
Let us differentiate x2 + (4x- 1)(x+2). Since we saw above that this is the sum of x2 and (4x- 1)(x+2), the first step is to use the rule for differentiating a sum (discussed in the preceding topic summary):

    d

    dx
    [x2 + (4x- 1)(x+2)] =
    d

    dx
    [x2]+
    d

    dx
    [(4x- 1)(x+2)]

Now we are left with two simpler functions to differentiate: x2, which is a power, so we use the power rule, and (4x- 1)(x+2), which is a product, so that we use the product rule on this:

    d

    dx
    [x2]+
    d

    dx
    [(4x- 1)(x+2)]
    = 2x+ 4(x+2) + (4x- 1)(1)

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Chain Rule

If f is a differentiable function of u and u is a differentiable function of x, then the composite f(u) is a differentiable function of x, and

    d

    dx
    [f(u)] = f'(u)
    du

    dx

Chain Rule In Words:

The derivative of f(quantity) is the derivative of , evaluated at that quantity, times the derivative of the quantity.

For instance, if f(u) = u0.5, then
    d

    dx
    [u0.5] = 0.5u- 0.5
    du

    dx

The derivative of a quantity raised to the 0.5 is 0.5 times the quantity raised to the -0.5, times the derivative of the quantity

Generalized Differentiation Rules
As the above Example illustrates, for every function of u whose derivative we know we now get a "generalized" differentiation rule:

Original Rule Generalized Rule
(Chain Rule)
d

dx
xn = nx n- 1
d

dx
un = nun- 1
du

dx
d

dx
4x- 1/2 = - 2x- 3/2
d

dx
4u- 1/2 = - 2u- 3/2
du

dx
d

dx
sin x = cos x
d

dx
sin u = cos u
du

dx
Examples

(First look over the generlized rules on the left.)

1.
d

dx
(1+x2)3 =
3(1+x2)2
d

dx
(1+x2)
=
3(1+x2)2,2x
=
6x(1+x2)2
2.
d

dx
2

(x+x2)3
=
d

dx
(x+x2)- 3
=
- 3(x+x2)- 4
d

dx
(x+x2)
=
- 3(x+x2)- 4 (1+2x)
=
−3(1+2x)

(x+x2)4
3.
d

dx
e(x+x2) =
e(x+x2)
d

dx
(x+x2)
=
e(x+x2)(1+2x)
4.
d

dx
(x- 2/x)- 1 =

Use proper graphing calculator format

Press here for an on-line tutorial on the chain rule.

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Derivatives of Logarithmic and Exponential Functions

The following table summarizes the derivatives of logarithmic and exponential functions, as well as their chain rule counterparts (that is, the logarithmic and exponential functions of a function).

Original Rule Generalized Rule
(Chain Rule)
d

dx
ln x   =
1

x
d

dx
ln u  =
1

u
du

dx
d

dx
logax=
1

x ln a
d

dx
logau =
1

u ln a
du

dx
d

dx
ex = ex
d

dx
eu = eu
du

dx
d

dx
ax = axln a
d

dx
au = auln a
du

dx
Examples

1.
d

dx
2 ln x =
2.
1

x
=
2

x
2.
d

dx
ln(2x) =
1

2x
d

dx
(2x)
=
2.
1

2x
=
1

x
3.
d

dx
log3(2x+1) =
1

(2x+1)ln 3
d

dx
(2x+1)
=
2.
1

(2x+1)ln 3
=
2

(2x+1)ln 3
4.
d

dx
ex2+1 =
ex2+1
d

dx
(x2+1)
=
2x ex2+1
5.
d

dx
2(x3+x) =

Use proper graphing calculator format

Press here for an on-line tutorial on derivatives of logarithmic and exponential functions.

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Derivatives of Trigonometric Functions

The following table summarizes the derivatives of the six trigonometric functions, as well as their chain rule counterparts (that is, the sine, cosine, etc. of a function).

Original Rule Generalized Rule
(Chain Rule)
d

dx
sin x = cos x
d

dx
sin u = cos u
du

dx
d

dx
cos x = - x
d

dx
cos u = - sin u
du

dx
d

dx
tan x = sec2 x
d

dx
tan u = sec2u
du

dx
d

dx
cotan x = - cosec2 x
d

dx
cotan u = - cosec2u
du

dx
d

dx
sec x = sec x tan x
d

dx
sec u = sec u tan u
du

dx
d

dx
cosec x
=- cosec x cotan x
d

dx
cosec u
=- cosec u cotan u
du

dx
Example

1.
d

dx
x sin x =
1.sin x + x cos x   Product rule
=
sin x + x cos x
2.
d

dx
cos(2x2+1) =
sin(2x2+1)
d

dx
(2x2+1)
=
sin(2x2+1).4x = 4x sin(2x2+1)
3.
d

dx
sec(x3) =
sec(x3) tan(x3)
d

dx
(x3)
=
sec(x3) tan(x3) . 3x2
=
3x2 sec(x3) tan(x3)
5.
d

dx
x cos(x2) =

Use proper graphing calculator format

Press here for on-line text on derivatives of trigonometric functions.

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Implicit Functions and Implicit Differentiation

Given an equation in x and y, we can think of y as an implicit function of x. We can find dy/dx without first solving for y as follows:

  • First, take the derivative with respect to x of both sides of the equation (treating y as "a quantity" in the chain rule).
  • Next, solve for dy/dx. This may give dy/dx in terms of both x and y.
  • To evaluate dy/dx at a specific value of x (or y), first substitute the given value in the original equation relating x and y to obtain a value for the other variable if necessary, and then substitute the values of x and y in the expression for dy/dx.

Logarithmic differentiation
This is the technique of first taking the (natural) logarithm of both sides of an equation and then finding dy/dx using implicit differentiation. Logarithmic differentiation is a useful alternative to the product and quotient rules when finding derivatives of particularly complicated expressions.

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Example

Suppose we want to find dy/dx given that

    xy+ y3x -ey = 4
First we take the derivative with respect to x (that is, d/dx) of both sides:
    d

    dx
    [xy + y3x -ey]=
    d

    dx
    [4]
    y + x
    dy

    dx
    + 3y2
    dy

    dx
    x + y3 - ey
    dy

    dx
    = 0
Now we solve for dy/dx by bringing together all terms with dy/dx on the left and taking the other terms to the right:
    x
    dy

    dx
    + 3y2
    dy

    dx
    x - ey
    dy

    dx
    =-y - y3
Now factor out dy/dx and then solve for it:
    dy

    dx
     x + 3xy2 - ey =-y - y3
    dy

    dx
    =
    -y - y3

    x + 3xy2 - ey

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Last Updated: April 2010
Copyright © Stefan Waner

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