A(x)= |4-3x|
A'(x)={ ,if x<4/3
,if x>4/3
I have to try to find the derivative of this function using a piecewise function and I'm a bit confused about how to do so. Could someone explain how?
Nevermind, I just figured out how to do it! My apologies!!
Since
|x| = x if x >= 0
|x| = -x if x < 0
|4-3x| > 0 if x < 4/3, so A(x) = 4-3x, and A' = -3
For x > 4/3, A(x) = -(4-3x), and A' = 3
To find the derivative of the function A(x) = |4-3x| using a piecewise function, we need to apply the definition of the derivative separately for different intervals of x:
1. For x < 4/3:
In this interval, the function |4-3x| simplifies to 3x-4, since the absolute value of 4-3x is x when x < 4/3. Therefore, A(x) = 3x - 4.
To find the derivative A'(x) for this interval, simply differentiate 3x - 4 with respect to x, treating the expression as a linear function. The derivative of a linear function is simply the coefficient of x, so A'(x) is equal to 3.
2. For x > 4/3:
For this interval, the function |4-3x| simplifies to -(3x-4), since the absolute value of 4-3x is -(3x-4) when x > 4/3. Therefore, A(x) = -(3x - 4).
Similarly, differentiate -(3x - 4) with respect to x. The derivative of a linear function with a negative sign simply changes the sign of the coefficient, so A'(x) is equal to -3.
Therefore, the piecewise derivative of the function A(x) = |4-3x| is:
A'(x) = 3, for x < 4/3
A'(x) = -3, for x > 4/3