Question

The function f(x)=ln∣x∣ has a piece-wise defined formula: ln∣x∣={lnx,ln(−x),x>0x<0. Use this piece-wise formula to compute f′(x).

Answer 1

The derivative of the function f(x) = ln|x| is f'(x) = 1/x. This result applies for both x > 0 and x < 0, using the chain rule for the negative case.

Step-by-step explanation:

To find the derivative of the function f(x) = ln|x|, we can use the piece-wise definition provided: for x > 0, f(x) = lnx, and for x < 0, f(x) = ln(-x). Since the natural logarithm function has a well-known derivative, we will apply it to both cases.

For x > 0:

• The derivative of f(x) = lnx is f'(x) = 1/x.

For x < 0:

• The derivative of f(x) = ln(-x) is f'(x) = -1/(-x) = 1/x, since the chain rule requires us to multiply the derivative of the natural logarithm function by the derivative of the inside function, which is -1, thereby negating the negative sign in the denominator.

Therefore, the derivative of f(x) = ln|x| can be summarized for all non-zero x as f'(x) = 1/x.

Note: At x = 0, the function f(x) is not defined, and thus its derivative at that point is also undefined.