On the basis of the assumption that the exponential function
is continuous everywhere and differentiable at 0, this function is differentiable everywhere and there is a formula for its derivative.
We can use a formula to find the derivative of
and the relationship
allows us to extend our differentiation formulas to include logarithms with arbitrary bases.
Logarithmic differentiation allows us to differentiate functions of the form
or very complex functions by taking the natural logarithm of both sides and exploiting the properties of logarithms before differentiating.
Key equations
Derivative of the natural exponential function
Derivative of the natural logarithmic function
Derivative of the general exponential function
Derivative of the general logarithmic function
For the following exercises, find
for each function.
the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement
A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.
A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike
Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.
the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon