

Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
Material Type: Exam; Professor: Jernigan; Class: Calculus I; Subject: Mathematics; University: Community College of Philadelphia; Term: Unknown 1989;
Typology: Exams
1 / 3
This page cannot be seen from the preview
Don't miss anything!
Math 171 Test 2 Solutions
Find the following derivatives. Simplify if possible, but don’t do anything silly.
d x dx
Chain Rule:
1 c cos 2 sin 2 sin
os x x x x
d x dx
d (^) x^2 e dx
Another Chain Rule:
2 2 2 2
x x e x xe
− − × − = −
d
dx
Produce Rule:
1 sin sin ln cos ln cos
x x x x x x x
× + × = + x
d x
dx x
Quotient Rule:
2 2
x x (^) x x
x x
2 x − 2
d (^) sin x x dx
Logarithmic Differentiation:
sin sin sin sin ln ln cos
x d^ x x x x x x x x dx x
Note that this was the answer to question #5.
sin 2
sin 2
d x
dx x
Same as Question #4 with x replaced by sin x. Therefore, by the Chain
Rule and the answer to #4:
2 2
cos sin 2 sin 2
cos x x x x
x f x =
1 10
x x
− ×
(i.e. why the power rule does not apply here.)
Because the variable is in the exponent!
x f x =
Logarithmic differentiation or memory:
x d x f x x dx
x ln
f x x
at a = 2
f = = f ′ = − = − ⇒ L x = + − x − = − + = 3 − − (^) −
x x
y y y y x y y x y y x x
1 2
tan ( ) 1
d x dx (^) x
− ⎡ ⎤ = ⎣ ⎦
and the chain rule to find
1 tan ( )
d (^) x e dx
− ⎡ ⎤ ⎣ ⎦
2 2
x x x x
e e e e
cos , , 2
sin ln
f x x g h x x (^) x
f g h x x
cos ln 2 2
x x x x x