If f is a differentiable function, find an expression for the derivative of each of the following functions

12a.  d) x⁶ f'(x) + 6x⁵ f(x)

Step-by-step explanation:

Use the multiplication formula for derivatives:

y = a · b       →     y' = a'b + ab'

y = x⁶ · f(x)      

a = x⁶          b = f(x)

a' = 6x⁵       b' = f'(x)

y' = a'b + ab'

y' = 6x⁵ f(x) + x⁶ f'(x)

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12b.  

Step-by-step explanation:

Use the division formula for derivatives:

      →    

Note: You can also move the denominator to the top (it will have a negative exponent) and use the multiplication formula for derivatives.

a 4/6 = 9/12   not proportional

step-by-step explanation:

to find which are not proportional, we can use cross products

if it is true, they are proportional.   if it is not true, they are not proportional.

a 4/6 = 9/12

4* 12 = 6*9

48 = 54

not this is not proportional  

b 2/3 = 12/18  

2* 18 = 3 * 12

36 = 36

this is proportional

c 5/4 = 10/8  

5*8 = 4*10

40 = 40

this is proportional

d 7/5 = 35/25

7*25 = 5 * 35

175 = 175

is proportional

the answer i believe is 27 units.

step-by-step explanation: