Derivatives-Second Part

Derivatives of Trigonometric Functions

Slope of y = cos x The curve y´ = –sin x as the graph of the slopes of the tangents to the curve y = cos x. Slope of y = cos x

Derivative of inverse sine y = sin-1x is equivalent to sin y = x Using implicit differentiation, y) x 1 cos y y = 1 If y = sin-1u Find the derivative for y = tan-1 u and sec-1u.

Using inverse cofunction identities, find the derivatives of the inverse cofunctions. The derivative of the inverse cofunction is the negative of the derivative of the function.

Derivatives of inverse trig functions

Arcsin function x = sin y The graph of y = sin–1 x has vertical tangents at x = –1 and x = 1.

Chain rule u turns 3 times as fast as x So y turns 3/2 as fast as x When gear A makes x turns, gear B makes u turns and gear C makes y turns., u turns 3 times as fast as x So y turns 3/2 as fast as x y turns ½ as fast as u Rates are multiplied

Outside/Inside method of chain rule derivative of inside derivative of outside wrt inside think of g(x) = u

Outside/Inside method of chain rule example derivative of inside inside derivative of outside wrt inside

Implicit Differentiation Although we can not solve explicitly for y, we can assume that y is some function of x and use implicit differentiation to find the slope of the curve at a given point y=f (x)

y2 is a function of y, which in turn is a function of x. If y is a function of x then its derivative is y2 is a function of y, which in turn is a function of x. using the chain rule: Find the following derivatives wrt x Use product rule

Higher Derivatives The derivative of a function f(x) is a function itself f ´(x). It has a derivative, called the second derivative f ´´(x) If the function f(t) is a position function, the first derivative f ´(t) is a velocity function and the second derivative f ´´(t) is acceleration. The second derivative has a derivative (the third derivative) and the third derivative has a derivative etc.

Find the second derivative for Find the third derivative for

Rate of change in radius of a sphere Examples of rates-assume all variables are implicit functions of t = time Rate of change in radius of a sphere Rate of change in volume of a sphere Rate of change in length labeled x Rate of change in area of a triangle Rate of change in angle,

The Waverley can reach its top speed in 5 minutes. During that time its distance from the start can be calculated using the formula D = t + 50t2 where t is the time in minutes and D is measured in metres. What is the Waverley’s top speed? Speed, v m/min, is the rate of change of distance with time. v = 1 + 100  5 = 501 m/sec How fast is it accelerating? Acceleration, a m/min/min, is the rate of change of speed with time.

I have no special talents. I am only passionately curious I have no special talents. I am only passionately curious. – Albert Einstein