Find the derivative of the function.

f(x) = x² ln(x)

f'(x) =

Find the derivative of the function.

f(x) = ln(e^8x)

f'(x) =

Find the derivative of the function.

f(x) = e⁷

f'(x) =

Find the derivative of the function.

f(x) = e^x ln(x²)

f'(x) =

For the function, find the indicated expressions.

f(x) =In(x)/x^4

(a)Findf'(x)

f'(x) =

(b)Find

f'(6).(Round your answer to five decimal places.)

f'(6) =

Find the derivative of the function.

f(x) = e^[(x3)3]

f'(x) =

If consumer demand for a commodity is given by the function below (where p is the selling price in dollars), find the price that maximizes consumer expenditure.

D(p) = 3000e^0.02p

p = $

The weekly sales (in thousands) of a new product are predicted to be

S(x) = 700 600e^0.3x afterxweeks. Find the rate of change of sales after the following amounts of time. (Round your answers to one decimal place.)

(a) 1 week thousand sales per week (b) 8 weeks thousand sales per week

A sum of $1000 at 11% interest compounded continuously will grow toV(t) = 1000e^0.11t dollars in t years. Find the rate of growth after the following amounts of time. (Round your answers to two decimal places.)

(a)0 years (the time of the original deposit) dollars per year

(b)10 years dollars per year

By calculating the first few derivatives, find a formula for the nth derivative of the function (k is a constant).

f(x) = e^kx

fⁿ(x) =

Find the second derivative of the function.

f(x) = e^x55

f''(x) =

Find the equation for the tangent line to the curve

y = f(x)

at the given x-value.

f(x) = e^{x3 1} at x = 1