Since the remaining four trigonometric functions may be expressed as quotients involving sine, cosine, or both, we can use the quotient rule to find formulas for their derivatives.
The derivative of the tangent function
Find the derivative of
Start by expressing
as the quotient of
and
Now apply the quotient rule to obtain
Simplifying, we obtain
Recognizing that
by the Pythagorean theorem, we now have
The derivatives of the remaining trigonometric functions may be obtained by using similar techniques. We provide these formulas in the following theorem.
Derivatives of
And
The derivatives of the remaining trigonometric functions are as follows:
Finding the equation of a tangent line
Find the equation of a line tangent to the graph of
at
To find the equation of the tangent line, we need a point and a slope at that point. To find the point, compute
Thus the tangent line passes through the point
Next, find the slope by finding the derivative of
and evaluating it at
Using the point-slope equation of the line, we obtain
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