CHAIN RULE

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This module will teach you the basics of direct and indirect proportions. These concepts will further help you in time and work questions.

Important Formulas – chain rule

• Direct Proportion

  Two quantities are said to be directly proportional, if on the increase or decrease of the one, the other increases or decreases the same extent.
  Examples

  1. Cost of the goods is directly proportional to the number of goods. (More goods, More cost)
  2. Amount of work done is directly proportional to the number of persons who did the work. (More persons, More Work)

• Indirect Proportion (inverse proportion)

  Two quantities are said to be indirectly proportional (inversely proportional) if on the increase of the one, the other decreases to the same extent and vice-versa.

Examples

• 1. Number of days needed to complete a work is indirectly proportional (inversely proportional) with the number of persons who does the work (More Persons, Less Days needed)
  2. The time taken to travel a distance is indirectly proportional (inversely proportional) with the speed in which one is travelling (More Speed, Less Time)

Solved Examples

Level 1

Answer : Option C

Explanation :

cost of x metres of wire = Rs. d

cost of 1 metre of wire = Rs.(d/x)

cost of y metre of wire = Rs.(y×d/x)=Rs. (yd/x)

Answer : Option B

Explanation :

Meal for 200 children = Meal for 120 men

Meal for 1 child = Meal for 120/200 men

Meal for 150 children = Meal for (120×150)/200 men=Meal for 90 men

Total mean available = Meal for 120 men

Renaming meal = Meal for 120 men – Meal for 90 men = Meal for 30 men

Answer : Option D

Explanation :
Let the required number of days be x

More men, less days (indirect proportion)

Hence we can write as

Men36:27}::x:18 ⇒36×18=27×x ⇒12×18=9×x

⇒12×2=x

⇒x=24

Answer : Option D

Explanation :

Let the number of revolutions made by the larger wheel be x

More cogs, less revolutions (Indirect proportion)

Hence we can write as

Cogs 6:14}: x: 21⇒6×21=14×x ⇒6×3=2×x ⇒3×3=x ⇒x=9

Answer : Option B

Explanation :

Let the required hours needed be x

More pumps, less hours (Indirect proportion)
More Days, less hours (Indirect proportion)

Hence we can write as

Pumps  3:4

::x:8

Days                      2:1

⇒3×2×8=4×1×x

⇒3×2×2=x

⇒x=12

Answer : Option D

Explanation :
Let the required number of days be x

More persons, less days (indirect proportion)
More hours, less days (indirect proportion)

Hence we can write as

Persons                39:30

::x:12

Hours    5:6
⇒39×5×12=30×6×x ⇒39×5×2=30×x ⇒39=3×x ⇒x=13

Answer : Option D

Explanation :

Let the required number of seconds be x

More cloth, More time, (direct proportion)

Hence we can write as

Cloth         0.128:25} :: 1:x

⇒0.128x=25 ⇒x=25/0.128 ⇒25000/128=3125/16≈195

Answer : Option A

Explanation :

15 cows ≡ 21 goats

1 cow ≡21/15 goats

35 cows ≡ (21×35)/15 goats≡(21×7)/3 goats≡7×7 goats ≡ 49 goats

Level 2

Answer : Option B

Explanation :

Assume that in x days, one cow will eat one bag of husk.

More cows, less days (Indirect proportion)
More bags, more days (direct proportion)
Hence we can write as

Cows    40:1         ::x:40

Bags     1:40

⇒40×1×40=1×40×x ⇒x=40

Answer : Option D

Explanation :
Let 200 gm potato costs x paise

Cost of ¼ Kg potato = 60 Paise
=> Cost of 250 gm potato = 60 Paise (∵ 1 Kg = 1000 gm => ¼ Kg = ¹⁰⁰⁰/₄ gm = 250 gm)

More quantity, More Paise (direct proportion)

Hence we can write as

Quantity  200:250} :: x:60

⇒200×60=250×x ⇒4×60=5×x ⇒4×12=x ⇒x=48

Answer : Option D

Explanation :

Persons worked = 104
Number of hours each person worked per day = 8
Number of days they worked = 30
Work completed = ²/₅

Remaining days = 56 – 30 = 26
Remaining Work to be completed = 1 – ²/₅ = ³/₅
Let the total number of persons who do the remaining work = x
Number of hours each person needs to be work per day = 9

More days, less persons(indirect proportion) More hours, less persons(indirect proportion)
More work, more persons(direct proportion)

Hence we can write as

Days     30:26

Hours    8:9                                   ::x:104

Work     35:25
⇒30×8×3/5×104=26×9×2/5×x

⇒x=(30×8×3/5×104)/(26×9×2/5)=(30×8×3×104)/(26×9×2)

=(30×8×104)/(26×3×2)=(30×8×4)/(3×2)=5×8×4=160

Number of additional persons required = 160 – 104 = 56

Answer : Option B

Explanation :
Let amount of work completed by y men working y hours per in y days = w units

More men, more work(direct proportion)
More hours, more work(direct proportion)
More days, more work(direct proportion)

Hence we can write as

Men                      x:y

Hours    x:y          ::x:w

Days                      x:y
⇒x³w=y³x ⇒w=y³x/x³=y³/x²

Answer : Option A

Explanation :
Let the required height of the building be x meter

More shadow length, More height (direct proportion)

Hence we can write as

Shadow length 40.25:28.75}:: 17.5:x

⇒40.25×x=28.75×17.5 ⇒x=(28.75×17.5)/40.25=(2875×175)/40250

= (2875×7)/1610=2875/230=575/46=12.5

Answer : Option A

Explanation :

Let the required price be x

More apples, More price (direct proportion)

Hence we can write as

Apples 357:(49×12)} :: 1517.25:x

⇒357x = (49×12)×1517.25⇒x = (49×12×1517.25)/357=(7×12×1517.25)/51

= (7×4×1517.25)/17

=7×4×89.25≈2500

Answer : Option D

Explanation :

Let required amount of coal be x metric tonnes

More engines, more amount of coal (direct proportion)

If 3 engines of first type consume 1 unit, then 1 engine will consume 1/3 unit which is its the rate of consumption.
If 4 engines of second type consume 1 unit, then 1 engine will consume 1/4 unit which is its the rate of consumption
More rate of consumption, more amount of coal (direct proportion)

More hours, more amount of coal(direct proportion)

Hence we can write as

Engines                                                                9:8

rate of consumption                       13:14                     ::24:x

hours                                                                    8:13
⇒9×1/3×8×x=8×1/4×13×24 ⇒3×8×x=8×6×13 ⇒3xX=6×13⇒x=2×13=26

Answer : Option A

Explanation :

Given that food was sufficient for 2000 people for 54 days
Hence, after 15 days, the remaining food was sufficient for 2000 people for 39 days (∵ 54 – 15 =39)
Let x number of people came after 15 days.
Then, total number of people after 15 days = (2000 + x)
Then, the remaining food was sufficient for (2000 + x) people for 20 days

More men, Less days (Indirect Proportion)⇒Men        2000:(2000+x)}  ::  20:39

⇒2000×39=(2000+x)20⇒100×39=(2000+x)⇒3900=2000+x⇒x=3900−2000=1900,

The chain rule is a formula for differentiating composite functions. It is used to find the derivative of a function that is composed of two or more other functions. The chain rule can be written in two different ways:

$$\dfrac{d}{dx}[w\Bigl(u(x)\Bigr)] = w’\Bigl(u(x)\Bigr)u'(x)$$

or

$$\dfrac{dy}{dx} = \dfrac{\partial y}{\partial u} \cdot \dfrac{du}{dx}$$

where $w$ and $u$ are any functions of $x$, and $y = w(u(x))$.

The chain rule can be used to find the Derivatives of many different types of functions. Some examples include:

• The derivative of a function of a function: $\dfrac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$
• The derivative of a composite function: $\dfrac{d}{dx}[w\Bigl(u(x)\Bigr)] = w’\Bigl(u(x)\Bigr)u'(x)$
• The derivative of a product of functions: $\dfrac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$
• The derivative of a quotient of functions: $\dfrac{d}{dx} \left[\dfrac{f(x)}{g(x)}\right] = \dfrac{g'(x)f(x) – f'(x)g(x)}{[g(x)]^2}$
• The derivative of a power function: $\dfrac{d}{dx}[x^n] = nx^{n-1}$
• The derivative of a logarithmic function: $\dfrac{d}{dx}[\ln(x)] = \dfrac{1}{x}$
• The derivative of an exponential function: $\dfrac{d}{dx}[e^x] = e^x$
• The derivative of a trigonometric function: $\dfrac{d}{dx}[\sin(x)] = \cos(x)$, $\dfrac{d}{dx}[\cos(x)] = -\sin(x)$, $\dfrac{d}{dx}[\tan(x)] = \sec^2(x)$, $\dfrac{d}{dx}[\cot(x)] = -\csc^2(x)$, $\dfrac{d}{dx}[\sec(x)] = \sec(x)\tan(x)$, $\dfrac{d}{dx}[\csc(x)] = -\csc(x)\cot(x)$

The chain rule is a powerful tool that can be used to find the derivatives of many different types of functions. It is an essential tool for calculus and is used in many different fields, including physics, chemistry, and engineering.

The chain rule can be derived using the definition of the derivative. The derivative of a function $f$ at a point $x$ is defined as:

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}$$

If $f$ is a composite function, then $f(x) = w(u(x))$ for some functions $w$ and $u$. We can write:

$$f(x+h) = w(u(x+h))$$

and

$$f(x) = w(u(x))$$

Subtracting these two equations, we get:

$$f(x+h) – f(x) = w(u(x+h)) – w(u(x))$$

Dividing both sides by $h$ and taking the limit as $h \to 0$, we get:

$$f'(x) = \lim_{h \to 0} \frac{w(u(x+h)) – w(u(x))}{h}$$

But $u(x+h) = u(x) + h$, so we can write:

$$f'(x) = \lim_{h \to 0} \frac{w(u(x) + h) – w(u(x))}{h}$$

We can factor out $w'(u(x))$ from the numerator:

$$f'(x) = \lim_{h \to 0} w'(u(x))

What is the derivative of a function?

The derivative of a function is a measure of how much the function changes as its input changes. It is calculated using the limit definition of the derivative, which is as follows:

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}$$

In other words, the derivative of $f$ at $x$ is the limit of the difference quotient $\frac{f(x+h) – f(x)}{h}$ as $h$ approaches zero.

What is the chain rule?

The chain rule is a formula that allows you to calculate the derivative of a composite function. A composite function is a function that is made up of two or more other functions. For example, the function $f(x) = (x^2 + 1)^3$ is a composite function of the functions $g(x) = x^2$ and $h(x) = x^3$.

The chain rule states that the derivative of a composite function is equal to the product of the derivatives of the inner and outer functions. In other words, if $f(x) = g(h(x))$, then $f'(x) = g'(h(x))h'(x)$.

What is the power rule?

The power rule is a formula that allows you to calculate the derivative of a power function. A power function is a function of the form $f(x) = x^n$, where $n$ is a real number.

The power rule states that the derivative of a power function is equal to $n$ times the function itself, multiplied by the power of $x$ minus one. In other words, if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

What is the product rule?

The product rule is a formula that allows you to calculate the derivative of the product of two functions. The product rule states that the derivative of the product of two functions $f$ and $g$ is equal to the sum of the derivatives of $f$ and $g$, multiplied by each other. In other words, if $f(x) = g(x)h(x)$, then $f'(x) = g'(x)h(x) + g(x)h'(x)$.

What is the quotient rule?

The quotient rule is a formula that allows you to calculate the derivative of the quotient of two functions. The quotient rule states that the derivative of the quotient of two functions $f$ and $g$ is equal to the difference of the derivatives of $f$ and $g$, multiplied by each other, divided by the square of $g$. In other words, if $f(x) = \frac{g(x)}{h(x)}$, then $f'(x) = \frac{g'(x)h(x) – g(x)h'(x)}{h(x)^2}$.

What is the implicit differentiation rule?

The implicit differentiation rule is a formula that allows you to differentiate an equation that is not explicitly written in terms of $x$. The implicit differentiation rule states that if $F(x, y)$ is an equation that is true for all values of $x$ and $y$, then $\frac{d}{dx} F(x, y) = \frac{\partial F}{\partial x} + \frac{\partial F}{\partial y} \frac{dy}{dx}$.

What is the logarithmic differentiation rule?

The logarithmic differentiation rule is a formula that allows you to differentiate an equation that contains logarithms. The logarithmic differentiation rule states that if $y = f(x)$, then $\ln y = \ln f(x) = \ln f(x)’ = \frac{d}{dx} \ln f(x) = \frac{f'(x)}{f(x)}$.

What is the exponential differentiation rule?

The exponential differentiation rule is a formula that allows you to differentiate an equation that contains exponential functions. The exponential differentiation rule states that if $y = f(x)$, then $\frac{d}{dx} e^{f(x)} = e^{f(x)}f'(x)$.

What is the trigonometric differentiation rule?

The trigonometric differentiation rules are a set of formulas that allow you to differentiate trigonometric functions. The trigonometric differentiation rules are as follows:

$$\frac{

1. The derivative of $e^x$ is:
   (a) $e^x$
   (b) $1$
   (c) $x$
   (d) $1+x$

2. The derivative of $\sin(x)$ is:
   (a) $\cos(x)$
   (b) $\sin(x)$
   (c) $-\sin(x)$
   (d) $-\cos(x)$

3. The derivative of $\cos(x)$ is:
   (a) $-\sin(x)$
   (b) $\sin(x)$
   (c) $\cos(x)$
   (d) $-\cos(x)$

4. The derivative of $x^n$ is:
   (a) $n x^{n-1}$
   (b) $n x^n$
   (c) $x^{n+1}$
   (d) $x^{n-1}$

5. The derivative of $\frac{1}{x}$ is:
   (a) $-\frac{1}{x^2}$
   (b) $\frac{1}{x^2}$
   (c) $x$
   (d) $-x$

6. The derivative of $e^{ax}$ is:
   (a) $a e^{ax}$
   (b) $e^{ax}$
   (c) $a^2 e^{ax}$
   (d) $a$

7. The derivative of $\sin(ax)$ is:
   (a) $a \cos(ax)$
   (b) $\cos(ax)$
   (c) $-a \sin(ax)$
   (d) $-\cos(ax)$

8. The derivative of $\cos(ax)$ is:
   (a) $-a \sin(ax)$
   (b) $-\cos(ax)$
   (c) $a \sin(ax)$
   (d) $\sin(ax)$

9. The derivative of $x^n \sin(ax)$ is:
   (a) $n x^{n-1} \sin(ax) + a x^n \cos(ax)$
   (b) $n x^{n-1} \sin(ax) – a x^n \cos(ax)$
   (c) $n x^{n-1} \cos(ax) + a x^n \sin(ax)$
   (d) $n x^{n-1} \cos(ax) – a x^n \sin(ax)$

10. The derivative of $\frac{1}{x} \sin(ax)$ is:
    (a) $-\frac{1}{x^2} \sin(ax) – a \frac{1}{x} \cos(ax)$
    (b) $\frac{1}{x^2} \sin(ax) + a \frac{1}{x} \cos(ax)$
    (c) $-\frac{1}{x^2} \cos(ax) – a \frac{1}{x} \sin(ax)$
    (d) $\frac{1}{x^2} \cos(ax) + a \frac{1}{x} \sin(ax)$