10.3 Derivative of Natural Logs
Chapter
Chapter 10
Section
10.3
Solutions 27 Videos

Distinguish between natural logarithms and common logarithms.

0.41mins
Q1

Determine the derivative.

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

0.28mins
Q3a

Determine the derivative.

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

0.22mins
Q3b

Determine the derivative.

 \displaystyle v = e^t\ln t 

0.59mins
Q3c

Determine the derivative.

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

0.37mins
Q3d

Determine the derivative.

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

0.30mins
Q3e

Determine the derivative.

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

1.03mins
Q3f

Differentiate.

 \displaystyle f(x) = x \ln x 

0.17mins
Q4a

Differentiate.

 \displaystyle y = e^{\ln x} 

0.21mins
Q4b

Differentiate.

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

0.27mins
Q4c

Differentiate.

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

0.43mins
Q4d

Differentiate.

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

1.06mins
Q4e

Differentiate.

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

1.48mins
Q4f

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

1.04mins
Q5a

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

0.55mins
Q5b

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

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

0.28mins
Q6a

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

Q6b

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

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

1.04mins
Q6c

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.

1.30mins
Q7a

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.

1.47mins
Q8

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

2.10mins
Q9a

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

1.10mins
Q10

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?

0.39mins
Q11a

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?

0.49mins
Q11bc

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?

0.54mins
Q11d

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).