What is the derivative of #f(x)=x*ln(x)# ?

Question

Christopher Allers · Accepted Answer

The function #f(x) = x* ln(x)# is of the form #f(x) = g(x) * h(x)# which makes it suitable for appliance of the product rule.

Product rule says that to find the derivative of a function that's a product of two or more functions use the following formula:

#f'(x) = g'(x)h(x) + g(x)h'(x)#

In our case, we can use the following values for each function:

#g(x) = x#

#h(x) = ln(x)#

#g'(x) = 1#

#h'(x) = 1/x#

When we substitute each of these into the product rule, we get the final answer:

#f'(x) = 1*ln(x) + x * 1/x = ln(x) + 1#

Learn more about the product rule here.

Evelyn Agard · Answer

undefined To find the derivative of $ f(x) = x \ln(x) $, you can use the product rule, which states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Applying this rule, we have:

$$ f'(x) = \frac{d}{dx}(x) \cdot \ln(x) + x \cdot \frac{d}{dx}(\ln(x)) $$

$$ f'(x) = 1 \cdot \ln(x) + x \cdot \frac{1}{x} $$

$$ f'(x) = \ln(x) + 1 $$

Therefore, the derivative of $ f(x) = x \ln(x) $ is $ f'(x) = \ln(x) + 1 $.