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)
.
Introduction to Derivative of ln x with Examples
Derivative of y = a^x
Derivative of y=\ln|x|
Derivative of Logarithmic Functions with Chain and Product Rules ex1
Derivative of Logarithmic Functions with Chain and Product Rules ex2
Derivative of Logarithmic Functions with Chain and Product Rules ex3
Derivative of Logarithmic Functions with Chain and Product Rules ex4
Definition of "e"