Deriving Special Functions

Question 0

(d/dx) cosx=

Answer 0

(d/dx) cosx= -sinx

Question 1

(d/dx) sinx=

Answer 1

(d/dx) sinx= cosx

Question 2

(d/dx) tanx=

Answer 2

(d/dx) tanx= (sec)^2 x

Because tanx= sinx/cosx
So (tanx)’= ___(sinx)’(cosx)-(sinx)(cosx)’__
(cos^2)x
= ___cosxcosx-sinx(-sinx)___
(cos^2)x
= ____(cos^2)x+(sin^2)x____
(cos^2)x
= 1/(cos^2)x
= (sec^2)x

Question 3

(d/dx) cscx=

Answer 3

(d/dx) cscx= -cscx*cotx

Question 4

(d/dx) secx=

Answer 4

(d/dx) secx= secx*tanx

Question 5

(d/dx) cotx=

Answer 5

(d/dx) cotx= -(csc)^2 x

Question 6

(d/dx) e^x=

Answer 6

(d/dx) e^x= e^x

Question 7

(d/dx) lnx=

Answer 7

(d/dx) lnx= 1/x

Question 8

(d/dx) log2(x)=

Answer 8

(d/dx) log2(x)= (1/x)/ln2= 1/(xln2)

Divide by the natural log of the base

Question 9

(d/dx) log(x)=

Answer 9

(d/dx) log(x)= 1/(xln10)

Remember log10(x) is written without the 10

Question 10

The product rule

Answer 10

Used for finding the derivative of the product of two functions

If y= thisthat,
Then y’= this’that + this*that’

Ex. For y= (x^3)(sinx)
y’= (x^3)’(sinx) + (x^3)(sinx)’
= (3x^2)(sinx) + (x^3)(cosx)

Question 11

The quotient rule

Answer 11

Used for finding the derivative of the product of two functions

If y= top/bottom’
Then y’= ___(top’bottom-topbottom’)___
(bottom)^2

Ex. For the derivative of y= sinx/(x^4)
      y'= \_\_\_(cosx)(x^4)-(sinx)(4x^3)\_\_\_
                                 x^8
         = \_\_\_\_(x^3)(xcosx-4sinx)\_\_\_\_
                              x^5

Question 12

The constant rule

Answer 12

For any number c, if f(x)= c, then f’(x)= 0

Because f(x)= c is a horizontal line

Question 13

The power rule

Answer 13

If f(x)= x^5
To find its derivative:
1. Take the power, 5, bring it in front of the x
2. Then reduce the power by 1 (in this example, the power becomes 4)
This gives you f’(x)= 5x^4

Works for any power: positive, negative, or a fraction

Question 14

Derivative of f(x)= x

Answer 14

f’(x)= 1

Power rule

Question 15

The constant multiple rule

Answer 15

Follows the power rule
Ex. f(x)= 4x^3
f’(x)= 12x^2

Question 16

The sum (and difference) rule

Answer 16

Ex. What’s f’(x)= x^6 + x^3 + x^2 + x + 10?
Use power rule
f’(x)= 6x^5 + 3x^2 + 2x + 1

Difference rule is the same but with minus signs

Question 17

Chain rule

Answer 17

For composite functions (functions inside other functions) ex. y= ✔️(4x^3 - 5)
1. Start with the outside function, ✔️, and differentiate that, IGNORING what’s inside.
So, y= ✔️(stuff)~ ✔️(x)= x^(1/2). Based on the power rule, y’= 1/2x^(-1/2)~ 1/2stuff^(-1/2)
2. Multiply the result from #1 by the inside function, stuff’. So, y’= 1/2stuff^(-1/2)*stuff’
3. Differentiate the inside stuff. Stuff’= 12x^2 by the power rule
4. Now plug in the real stuff and it’s derivative where they belong. y’= 1/2(4x^3-5)^(-1/2) * 12x^2
5. Simplify. 6x^2 * (4x^3-5)^(-1/2)
____6x^2_____
(4x^3-5)^(1/2) OR
____6x^2_____
✔️(4x^3-5)

Question 18

Chain rule for a third composite function

Answer 18

y= sin^3(5x^2 - 4x)
1. Rewrite the cubed sine function:
y= [sin((5x^2 - 4x)]^3
Now the innermost function is in the innermost parentheses (5x^2 - 4x), the next function is sin(stuff), and the third is stuff^3
2. Outermost function: stuff^3 and its derivative is given by the pore rule: 3stuff^2
3. Now multiply that by stuff’: 3stuff^2 * stuff’
4. Plug sin(5x^2 - 4x) back into “stuff”
3[sin(5x^2 - 4x)]^2 * [sin(5x^2 - 4x)]’
5. Use chain rule again. You have to treat
sin(5x^2 - 4x) as if it were the original problem and take its derivative.
Derivative of sinx= cosx so derivative of sin(stuff)= cos(stuff) * stuff’
6. Plug 5x^2 + 4x into stuff
Cos(5x^2 + 4x) * (10x-4)
7. Put it all together:
3[sin(5x^2 - 4x)]^2 * Cos(5x^2 + 4x)*(10x-4)
8. Simplify:
(30x-12) * sin^2(5x^2 - 4x) * Cos(5x^2 + 4x)

Question 19

What to do if you’re not sure where to begin differentiating a complex function

Answer 19

Ex. Differentiating 4x^2sin(x^3), which involves the chain rule and product rule.
1. Imagine plugging a number into x and then evaluating the expression one step at a time

Question 20

Differentiating implicitly

Answer 20

Ex. 5y^4 * y' + 6x= cosx - 12y^2 * y'
1. Differentiate each term on both sides of the equation 
 2. Collect all terms containing a y' on the left side of the equation and all other terms on the right side of the equation
     5y^4 * y' + 12y^2 * y'= cosx - 6x
3. Factor out y'
    y'(5y^4 + 12y^2)= cosx - 6x
4. Divide for the final answer 
    y'= \_\_\_cosx - 6x\_\_\_\_
           (5y^4 + 12y^2)

Question 21

Logarithmic differentiation

Answer 21

Ex. f(x)= (x^3 -5)(3x^4 + 10)(4x^2 - 1)(2x^5 - 5x^2 + 10)
1. Take the natural log of both sides
ln f(x)= ln [(x^3 -5)(3x^4 + 10)(4x^2 - 1)(2x^5 - 5x^2 + 10)]
2. Use the property for the log of a product
ln f(x)= ln(x^3 -5) + ln(3x^4 + 10) + ln(4x^2 - 1) + ln(2x^5 - 5x^2 +10)
3. Differentiate both sides, and according to the chain rule, the derivative of ln f(x) is 1/f(x) * f’(x) OR f’(x)/f(x)– so basically use the chain rule
f’(x)/f(x)= 3x^2/(x^3 -5) + 12x^3/(3x^4 + 10) + 8x/(4x^2 - 1) +
(10x^4 - 10x)/(2x^5 - 5x^2 +10)
4. Multiply both sides by f(x)
f’(x)= (x^3 -5)(3x^4 + 10)(4x^2 - 1)(2x^5 - 5x^2 +10)

Question 22

Finding a second, third, fourth, etc. derivative

Answer 22

The second derivative of a function is the derivative of its first derivative, and the third derivative of a function is the derivative of its second derivative... And all are obtained by the power rule
Ex. f(x)= x^4 - 5x^2 + 12x - 3
      f'(x)= 4x^3 - 10x + 12
      f''(x)= 12x^2 - 10
      f'''(x)= 24x
      f''''(x)= 24
      f'''''(x)= 0
      etc. = 0