Marginal Cost Businesses can buy multiple licenses for data compression software at a total cost of approximately C(x) = 96x2/3 dollars for x licenses. Find the derivative of this cost function at the following values x=8 and x=64
I do not know where to start?
Start where the instructions say "find the derivative"!
just use the power rule
C'(x) = 96 * 2/3 x^(-1/3) = 64/∛x
C'(8) = 64/∛8 = 64/2 = 32
C'(64) = 64/∛64 = 64/4 = 16
To find the derivative of the cost function, you need to use the power rule of differentiation. The power rule states that if you have a function of the form f(x) = ax^n, where "a" and "n" are constants, then the derivative of f(x) is given by f'(x) = anx^(n-1).
In this case, the cost function is given by C(x) = 96x^(2/3). To find the derivative of C(x), you can follow these steps:
Step 1: Rewrite the cost function using fractional exponents: C(x) = 96 * x^(2/3).
Step 2: Apply the power rule to find the derivative:
C'(x) = d/dx (96 * x^(2/3))
= 96 * (2/3) * x^((2/3) - 1)
= 64 * x^(-1/3).
So, the derivative of the cost function is C'(x) = 64 * x^(-1/3).
Now, you can find the derivative of the cost function at x = 8 and x = 64 by substituting these values into the derivative expression:
At x = 8:
C'(8) = 64 * 8^(-1/3)
= 64 / (2)
= 32.
At x = 64:
C'(64) = 64 * 64^(-1/3)
= 64 / 4
= 16.
Therefore, at x = 8, the derivative is 32, and at x = 64, the derivative is 16.
To find the derivative of the cost function C(x) = 96x^(2/3), you can use the power rule of differentiation, which states that for a function of the form f(x) = ax^n, the derivative is given by f'(x) = anx^(n-1).
In this case, the constant coefficient is 96, and the exponent is 2/3. Applying the power rule, we have:
C'(x) = (2/3) * 96 * x^((2/3) - 1)
Simplifying further, we get:
C'(x) = 64x^(-1/3)
Now that we have the derivative of the cost function, we can find the derivative at specific values of x.
For x = 8:
C'(8) = 64(8)^(-1/3)
= 64/2
= 32
For x = 64:
C'(64) = 64(64)^(-1/3)
= 64/4
= 16
Therefore, the derivative of the cost function at x = 8 is 32 and at x = 64 is 16.