Distinguish between natural logarithms and common logarithms.

Determine the derivative.

\displaystyle y = \ln(5x + 8)

Determine the derivative.

\displaystyle y = \ln(x^2 + 1)

Determine the derivative.

\displaystyle v = e^t\ln t

Determine the derivative.

\displaystyle v = \ln \sqrt{x + 1}

Determine the derivative.

\displaystyle s = \ln (t^3 -2t^2 + 5)

Determine the derivative.

\displaystyle w = \ln \sqrt{z^2 + 3z}

Differentiate.

\displaystyle f(x) = x \ln x

Differentiate.

\displaystyle y = e^{\ln x}

Differentiate.

\displaystyle y = e^{x} \ln x

Differentiate.

\displaystyle g(x) = \ln(e^{-x} + x e^{-x})

Differentiate.

\displaystyle s = \ln(x^3 - 2x^2 + 5)

Differentiate.

\displaystyle y = \ln \sqrt{x^2 + 3x}

If g(x) = e^{2x -1}\ln(2x - 1), evaluate g'(1).

If f(x) = \ln(\frac{x - 1}{3x + 5}), evaluate f'(5).

Solve the equation for f'(x) = 0:

f(x) = \ln(x^2 + 1)

Solve the equation for f'(x) = 0:

Solve the equation for f'(x) = 0:

f(x) = (x^2 + 1)^{-1}\ln(x^2 + 1)

Determine the equation of the tangent to the curve defined by f(x) = \frac{\ln \sqrt[3]{x}}{x} at the point where x = 1.

Determine the equation of the tangent to the curve defined by y = \ln x - 1 that is parallel to the straight line with equation 3x - 6y - 1 = 0.

If f(x) = (x\ln x)^2, determine al the points at which the graph of f(x) has a horizontal tangent line.

Determine the equation of the tangent to the curve defined by y = \ln(1 + e^{-x}) at the point where x = 0.

The velocity, in kilometres per hour, of a car as it begins to slow down is given by the equation v(t) = 90- 30\ln(3t + 1), where t is in seconds.

What is the velocity of the car as the driver beings to brake?

The velocity, in kilometres per hour, of a car as it begins to slow down is given by the equation v(t) = 90- 30\ln(3t + 1), where t is in seconds.

i. What is the acceleration of the car?

ii. What is the acceleration at t = 2?

The velocity, in kilometres per hour, of a car as it begins to slow down is given by the equation v(t) = 90- 30\ln(3t + 1), where t is in seconds.

How long does the car take to stop?

Use the definition of the derivative to evaluate \displaystyle \lim_{h \to 0} \frac{\ln(2 + h) - \ln 2}{h}

Consider f(x) = \ln(\ln x).

a. Determine f'(x).

b. State the domains of f(x) and f'(x).