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AP Calculus

Chain Rule, Deriving Trig Functions and Position Notes

Name___________________________________

Date________________

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-1-

Differentiate each function with respect to the given variable.

1) h = 5t2 + 2

3t4 + 5

2) g = (-s2 - 5)-33s3 - 5

3) h(s) = (4s3 + 1)-3

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-2-

4) f = ((3x + 5)3 - 2)

1

4

Differentiate each function with respect to x.

5) f (x) = csc 4x4

cot x3

6) f (x) = cot (sin 3x3)

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-3-

7) y = -16x

(2x - 5)2

For each problem, find the equation of the line tangent to the function at the given point.

8) y = -16x

(2x - 5)2 at x = 2

For each problem, find the equation of the line normal to the function at the given point.

9) y = -sec (x) at x = -p

6

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-4-

A particle moves along a horizontal line. Its position function is s(t) for t ³ ³ 00. For eachproblem, find the velocity function v(t), the acceleration function a(t), the times t when theparticle changes directions, the intervals of time when the particle is moving left and movingright, the times t when the acceleration is 0, and the intervals of time when the particle isslowing down and speeding up.

10) s(t) = -t3 + 28t2 - 196t

You are given a table containing some values of differentiable functions f (x), g(x) and theirderivatives. Use the table data and the rules of differentiation to solve each problem.

11) x f (x) f '(x) g(x) g'(x)1 5 -1 1 2

2 4 -1 33

2

3 3 -3

24

3

2

4 1 -1

26 0

5 2 1 4 -3

2

6 3 1 3 -1

Part 1) Given h1(x) = ( f (x))2, find h

1'(3)

Part 2) Given h2(x) = f (g(x)), find h

2'(3)