3.5: Derivatives of Trig Functions Objective: Students will be able to find and apply the derivative of a trigonometric function.

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3.5: Derivatives of Trig Functions Objective: Students will be able to find and apply the derivative of a trigonometric function.

y = sin x SKETCH A GRAPH OF THE DERIVATIVE OF Y = SIN X. WHAT DO YOU NOTICE?

y = cos x SKETCH A GRAPH OF THE DERIVATIVE OF Y = COS X. WHAT DO YOU NOTICE?

Derivative of sinx and cosx:

EXAMPLES: Find the derivative. 1. y=x 2 + 2cosx 2. y=2sinxcosx 3.

Use the quotient and reciprocal identities to find the derivative of the other 4 trig functions.

Derivatives of Other 4 Trig Functions **Notice: All trig functions that begin with a “c” have a negative derivative

EXAMPLES: 1. y= -x + tanx 2. y=5 θsecθ + θtanθ 3. y=-cscx-sinx

Other examples: 1.

Find f’( ) given

Do not use a calculator: What is ? (a) 1 (b) (c) 0 (d) -1 (e) The limit does not exist

Find the equation of the tangent line and the normal line of f(x) = tan(x) at the point

Write the equation of the tangent line to the curve y=x 2 cosx at x=2.

Application of the 3 rd Derivative A sudden change in acceleration is a “jerk”. Jerk = j(t) = a’(t)= v’’(t)= s’’’(t) Example: Find the velocity, acceleration and jerk of an object whose position function is s(t)=36t 2 +t 3, (0 < t < 6)