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DERIVATIVES OF TRIGONOMETRIC FUNCTIONS. When we talk about the function f defined for all real numbers x by , it is understood that means the sine of the angle whose radian measure is . A similar convention holds for the other trigonometric functions cos , tan, csc , sec, and cot.
When we talk about the function f defined for all real numbers x by
, it is understood that means the sine of
the angle whose radian measure is .
A similar convention holds for the other trigonometric functions
cos, tan, csc, sec, and cot
are positive so we can write
Divide it by
Take the inverse
By the Squeeze Theorem, we have:
However, the function (sin )/ is an even function.
So, its right and left limits must be equal.
Hence, we have:
Using the same methods as in the case of finding derivative of , we can prove:
The tangent function can also be differentiated by using the definition of a derivative.
However, it is easier to use the Quotient Rule together with formulas for derivatives of as follows.
The derivatives of the remaining trigonometric functions — csc, sec, and cot — can also be found easily using the Quotient Rule.
For what values of x does the graph of f have a horizontal tangent?
How to differentiate composite function
It turns out that the derivative of the composite
function f ◦ g is the product of the derivatives of f and g.
Proof goes over the head, so forget about that.
This fact is one of the most important of the differentiation rules.
It is called the Chain Rule.
If u changes twice as fast as x and y changes three times as fast as u, it seems reasonable that y changes six times as fast as x.
So,we expect that:
The Chain Rule can be written either
in the prime notation
if y = f(u) and u = g(x), in Leibniz notation:
easy to remember because, if dy/du and du/dx were quotients, then we could cancel du.
However, remember: du/dx should not be thought of as an actual quotient
Let’s go back:
In order not to make your life too complicated (it is already enough), we’ll introduce one way that is most common and anyway, everyone ends up with that one: Leibnitz
Differentiate y = (x3 – 1)100
Taking u= x3 – 1 and y = u100
Find f’ (x) if
Find the derivative of
The reason for the name ‘Chain Rule’ becomes clear when we make a longer chain by adding another link.
Suppose that y = f(u), u = g(x), and x = h(t),
where f, g, and h are differentiable functions,
then, to compute the derivative of y with
respect to t, we use the Chain Rule twice:
The slope of a parametrized curve is given by:
The chain rule enables us to find the slope of parametrically defined curves x = x(t) and y = y(t):