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Find the derivative.
    y = 17x^-2 – 6x³ + 13x
 

 
n^{n – 1}(17x^-2 – 6x³ + 13x)

-2(17x^{-2 – 1}) – 3(6x^{3 – 1}) + 13(x^{1 – 1}) =         -34x^-3 -18x² + 13
 
This online calculator will solve it:
https://www.derivative-calculator.net/
   
Find the derivative.
f(x) =  -  -  find f’(x)
 

   Find the derivative.  f(x) =  -  -      find f’(x)
 
6 ·  -  -
 
6 ·  x^{-1/2 – 1} +  - ^-6 =
 

 
This calculator will find the answer for you
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Enter in the equation like so:
 

   
The total cost to produce x handcrafted wagons is C(x) = 60 + 4x – x² + 8x³ Find the marginal cost when x equals 6.
 

 
C(x) = 60 + 4x – x² + 8x³
First, we rearrange the equation in the order of proper operation = 8x³ – x² + 4x + 60
Then we find the derivative of 8x³ – x² + 4x + 60, and then plug in the value for x = 6
n^{n – 1}(8x³ – x² + 4x + 60)
It is easiest to split this up like so: *Remember, n = the exponents.
 
3 · 8x^{3 – 1} = 24x²
2 · -x^{2 – 1} = -2x
1 · 4x^{1 – 1} = 4
60 · 0 = 0
 
Now put the equation back together
derivative = 24x² – 2x + 4
 
and now plug in the value for x = 6
24(6)² – 2(6) + 4 =
864 – 12 + 4 = 856
 
This calculator will find the derivative for you.
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Enter in the equation like so:
*Remember, you still need to plug in the value for x = 6 after.

   
If the price of a product is given by P(x) =    + 1400 where x represents the demand for the​ product,  find the rate of
change of price when the demand is 32.
 

 
We need to find the derivative of    and plug in the given value of 32 however,
all we really need to do is write this equation and solve -1,024 ÷ 32 ÷ 32 = -1.
 
Let’s do it the long way for practice.
Solve the long way:
We need to find the derivative of
and plug in the value for the demand 32.
The derivative = -
Now plug in the value for the demand (32).
 
 = Answer = -1
 
The derivative = -
Now plug in the value for the demand (32).
 
 = -1
 
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Use the product rule to find the derivative.
F(x) = (2x – 5) (3x³ – x² + 1)
 

 
We need to factor the expression and find the derivative with the product rule.
It is easiest to break the equations up and then put it back together and simplify.
This is a little long, so you need to be careful.
 
Solve the long way:
2x(3x³ – x² + 1) = 6x⁴ – 2x³ + 2x
-5(3x³ – x² + 1) = -15³ + 5x² - 5
Now place them in order and Simplify         6x⁴ (-2x³ -15³) + 5x² + 2x – 5   simplify           6x⁴ – 17x³ + 5x² + 2x – 5
Now find the derivative - Solve the long way: Find the derivative of 6x⁴ – 17x³ + 5x² + 2x – 5
 
n^{n – 1} (6x⁴ – 17x³ + 5x² + 2x – 5)
*Remember n = the exponents.
 
4 · 6x^{4 – 1} = 24x³
3 · -17x^{3 – 1} = -51x²
2 · 5x^{2 – 1} = 10x
2x = 2
0 · -5 = 0
Now put the equation back together.
24x³ – 51x² + 10x + 2
 
This calculator will find the derivative for you.
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Use the quotient rule to find the derivative.
f(x) =
 

Solve the long way

All we need to do is square the denominator x⁷ + 2 and move and multiply the numerator by x⁷ + 2
So, we have  =

Now all we need to do is find the derivative of the numerator x⁷ + 2, which = 7x⁶
The answer =
 
This calculator will find the derivative for you.
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You need to enter EXACTLY as shown or it will not be solved.
 

   
The total cost to produce x units of perfume is C(x) = (6x + 7) (3x + 3).
Find the marginal average cost​ function, C’(x)
 

 
After I solved about a dozen of these calculus problems, I solved them like below, and they were all correct,
So, if you feel confident enough, you can do the same. I simply followed the steps below.
6x(3x) + 7(3) = 18x² + 21
Divide 21 by x², subtract from the fraction, and leave 18 alone on the left.
However, the long way is below.

And in this case, we would need to find the derivative which will be explained below.
 
First, we need to solve for (6x + 7) (3x + 3)
6x (3x + 3) = 18x² + 18x
7 (3x + 3) = 21x + 21
18x² + (18x + 21x) + 21 =
18x² + 39x + 21
Now we need to write the equation by dividing by x and find the derivative.
 
now simplify as so.
 = 18x,  = 39,  =  (we do not move the x up)
 
Now we put the equation back together and find the derivative.
Find the derivative of: 18x + 39 +
18x = 18
39 = 0
 
 =
How to find the derivative of !
We square the denominator.
Next we move x up, and the fraction is then negative so we have

Next we do the same as we did above. We find the derivative of the numerator 21x which is 21
The answer to this part of the equation =
Now we put the equation back together.

 
https://www.derivative-calculator.net/
You need to enter EXACTLY as shown or it will not be solved.

  The demand function for a certain product is given by D(p) =
Find the marginal demand D’(p)
 

 
There are a couple of ways to solve this problem, but we will do it the simplest way.
The quotient rule is:
 
(  =
 
In this case,
f = 3p +180
g = 6p + 15
 
So, the easiest way is to write the equation as so and factor.
Pay close attention to how the numbers are swapped and
*g is squared on the bottom of the fraction (and will remain so).

Now factor the top
18p + 45 – 18p + 1080
18p – 18p = 0
45 – 1080 = -1,035
Now put the equation back together.
 

 
This is the way the book shows how to solve these problems,
However, the above is easiest and less confusing.
 

   
Find the derivative.
 
 
 

The shortest way to do this is the same as we have been doing (fractions)
Since it is a square root, it will be a fraction. We will start with .
Next, we multiply the numerator by the denominator without the square root. We have
Now we square the denominator, like we have done above, however, the 2 will be on the left side of the square root.
So, we have  and then we find the derivative of the numerator which = 4.
Now we have . We can now simply and remove the square (left of the square root) by dividing 4 / 2
We are left with the answer =
 
https://www.derivative-calculator.net/
You need to enter EXACTLY as shown or it will not be solved.
 

 
   
Find the derivative.
 
      
 

 
https://www.derivative-calculator.net/
You need to enter EXACTLY as shown or it will not be solved.
 

 
Or solve the long way.
 

 
 

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