Find f'(7) for f(x)=6x^ln(x)

Use chain rule.

Let
y=6xln(x)
take log on both sides:
ln(y) = ln(6)+ln(x)*ln(x)
ln(y) = ln(6) + ln²(x)
differentiate both sides w.r.t. x
(1/y)y' = 0 + 2ln(x)*(1/x)
y' = dy/dx
= y*2ln(x)*(1/x)
= 6xln(x)*2ln(x)*(1/x)
= 12xln(x)-1*ln(x)

Check my calculations.

To find f'(7) for the function f(x) = 6x^ln(x), we need to take the derivative of f(x) and then evaluate it at x = 7.

Let's start by finding the derivative of f(x):

f'(x) = d/dx (6x^ln(x))

To do this, we will use the logarithmic differentiation technique:

1. Take the natural logarithm of both sides of the equation:
ln(f(x)) = ln(6x^ln(x))

2. Apply the logarithmic properties to simplify the equation:
ln(f(x)) = ln(6) + ln(x) ln(x)

3. Differentiate both sides of the equation implicitly with respect to x:
(1/f(x)) * f'(x) = 0 + 1/x * 1 + ln(x) * 1
(1/f(x)) * f'(x) = 1/x + ln(x)

4. Multiply both sides of the equation by f(x):
f'(x) = f(x) * (1/x + ln(x))

Now we have the derivative f'(x) in terms of f(x), x, and natural logarithm. Next, we can evaluate f'(7) by substituting x = 7:

f'(7) = f(7) * (1/7 + ln(7))

To compute f(7), plug in the value of x into the original function:

f(7) = 6 * 7^ln(7)

Calculating ln(7) is a bit tricky as it's an irrational number, but many calculators or computer programs have a built-in natural logarithm function.

Finally, substitute the value of f(7) into the equation:
f'(7) = f(7) * (1/7 + ln(7))

Now you can evaluate f'(7) by calculating 7^ln(7) using a calculator or computer program, and then substitute the value into the equation above to calculate f'(7).