{Chapter 4 Practice

AP Calculus

Differentiate:𝑑𝑑𝑥 sec 𝑥=¿

𝑑𝑑𝑥 csc𝑥=¿

𝑑𝑑𝑥 tan 𝑥=¿

𝑑𝑑𝑥 cot𝑥=¿

𝑑𝑑𝑥 sin

−1𝑥=¿

𝑑𝑑𝑥 sec

−1 𝑥=¿

𝑑𝑑𝑥 csc

−1𝑥=¿

𝑑𝑑𝑥 tan

− 1𝑥=¿

Differentiate: sec x tan x

-csc x cot x

𝑑𝑑𝑥 tan 𝑥=𝑠𝑒𝑐2𝑥

𝑑𝑑𝑥 cot𝑥=−𝑐𝑠𝑐2𝑥

𝑑𝑑𝑥 sin

−1𝑥=1

√1−𝑥2

𝑑𝑑𝑥 csc

−1𝑥=− 1¿𝑥∨√𝑥2−1

𝑑𝑑𝑥 tan

− 1𝑥=1

𝑥2+1

To the nearest thousandth, calculate the slope of the tangent where x = 4:

𝑥2−4 𝑦2=4

To the nearest thousandth, calculate the slope of the tangent where x = 4:

𝑥2−4 𝑦2=4Differentiate implicitly:

2 𝑥 ∙ 𝑑𝑥𝑑𝑥 −8 y ∙𝑑𝑦𝑑𝑥=0

¿−8 y ∙𝑑𝑦𝑑𝑥=−2 𝑥

¿−8 y ∙𝑑𝑦𝑑𝑥=−2𝑥−8 𝑦→

𝑑𝑦𝑑𝑥=

𝑥4 𝑦

Find coordinates of y when x = 4and substitute into dy/dx equation:

h𝑊 𝑒𝑛𝑥=4 , 𝑦=±√3𝑑𝑦𝑑𝑥 =

44¿¿

𝑑𝑦𝑑𝑥 =

44¿¿

• Be prepared for NO CALCULATOR section!• Basic chain rule, product rule, quotient

rule• Basic trig derivatives• Inverse trig derivatives• Implicit differentiation • Use limits to find values to make a

piecewise function differentiable (and continuous).

• Related Rates

Ch. 4 Test Review Topics

Derivatives Practice

𝑑𝑑𝑥 𝑐𝑜𝑠

4(5 𝑥−4 )

𝑑𝑑𝑥

7𝑐𝑜𝑠5𝑥

𝑑𝑑𝑥 x ∙ csc

−15𝑥

𝑑𝑑𝑥

𝑒3𝑥7 𝑥−4

Derivatives Practice

𝑑𝑑𝑥 𝑐𝑜𝑠

4 (5𝑥−4 )=−20 (cos (5 𝑥−4 ) )3 ∙ sin (5 𝑥−4 )

= 35 sec 5x tan 5x

=

=

Differentiate implicitly:

4 𝑥2+ tan 𝑥𝑦=𝑦3

Differentiate implicitly:

Derivative:

Useful Related Rates Formulas

𝐶𝑜𝑛𝑒𝑉𝑜𝑙𝑢𝑚𝑒 :𝑉=13 𝜋 𝑟

2h

𝐶𝑖𝑟𝑐𝑙𝑒 𝐴𝑟𝑒𝑎 :𝜋𝑟2

Cylinder Volume: V =

h𝑃𝑦𝑡 𝑎𝑔𝑜𝑟𝑒𝑎𝑛 h𝑇 𝑒𝑜𝑟𝑒𝑚 :𝑎2+𝑏2=𝑐2

𝑆𝑖𝑚𝑖𝑙𝑎𝑟 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 (𝑆𝑒𝑡 𝑢𝑝𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛)

Ch. 4 R Problems, pg. 180: R4 ad, R5a, R6, R8b, R9 (pretty hard)

Suggested Review

Additional ReviewOnline videos, PPTSExamples from notes4.2 #1-15 odd, 4.3 1-19 odd4.4 1-25 odd, 4.5 13-23 odd