Determine the derivative of the function ƒ(x) = x^3 − 12x^2. Now determine the derivative of the function g(x) = x^3 − 12x^2 + 6. How are the two functions related? How are the derivatives of the two functions related?
I know the derivative, however how do i explain how the functions are related except that they are the exact same but one has a vertical translation up 6 units?
That is in fact how they are related.
And since the graphs are identical, the derivatives are the same.
To determine the derivative of a function, you can follow a basic process called differentiation. The derivative of a function represents its rate of change, or how it is changing with respect to its independent variable (in this case, x). Here's how you can find the derivatives of the given functions:
1. For the function ƒ(x) = x^3 − 12x^2:
To find the derivative of this function, you can apply the power rule of differentiation. According to the power rule, when you have a term of the form x^n, the derivative of that term is nx^(n-1).
So, differentiating each term of ƒ(x) separately, we have:
ƒ'(x) = d/dx (x^3) - d/dx (12x^2)
= 3x^2 - 24x
2. For the function g(x) = x^3 − 12x^2 + 6:
To find the derivative of g(x), you need to differentiate each term similarly, as we did for ƒ(x). Since the derivative of a constant (like 6) is zero, the last term does not affect the derivative.
g'(x) = d/dx (x^3) - d/dx (12x^2) + d/dx (6)
= 3x^2 - 24x + 0
= 3x^2 - 24x
Now, let's discuss how the two functions are related. Both functions, ƒ(x) and g(x), have the same derivative, which is 3x^2 - 24x. The only difference between them is the additional constant term in g(x). This means that g(x) is a vertical translation of 6 units upward from ƒ(x).
In other words, we can say that g(x) is obtained by adding a constant value (in this case, 6) to ƒ(x). This constant term does not affect the derivative because the derivative of a constant is always zero. Hence, the derivatives of the two functions are identical, showing that the derivative only captures information about the slope of the function and not its vertical positioning.