Review of Derivatives

Question 1

The derivative of ƒ @ a:

Answer 1

lim (ƒ(a+h)−ƒ(a))/((a+h)−a) as h->0 or, lim Δƒ(a)/Δa as h->0

Question 2

Differential operator notation:

Answer 2

d/dx(ƒ(x))=ƒ′(x)

Question 3

d/dx(c)=?

Answer 3

0, for any constant c

Question 4

d/dx(ƒ(x)±g(x))=?

d/dx(u±v)=?

Answer 4

d/dx(ƒ(x))±d/dx(g(x))=ƒ′(x)±g’(x) *differentiation of sums is term-by-term!

u’±v’

Question 5

d/dx(xⁿ)=?

Answer 5

nxⁿ⁻¹, for any real n

Question 6

d/dx(kƒ(x))=?

Answer 6

kd/dx(ƒ(x))=kƒ′(x), for any real k and differentiable ƒ

Question 7

d/dx(sinx)=?

Answer 7

cosx

Question 8

d/dx(cosx)=?

Answer 8

−sinx

Question 9

d/dx(e^x)=?

Answer 9

e^x

Question 10

d/dx(ƒ(x)g(x))=?

d/dx(uv)=?

Answer 10

ƒ′(x)g(x)+ƒ(x)g’(x)

u’v+uv’

Question 11

d/dx(ƒ(x)/g(x))=?

d/dx(u/v)=?

Answer 11

(ƒ′(x)g(x)−ƒ(x)g’(x))/((g(x))^2)

(v’u−uv’

Question 12

d/dx(tanx)=?

Answer 12

sec^2x

Question 13

d/dx(cotx)=?

Answer 13

−csc^2x

Question 14

d/dx(secx)=?

Answer 14

secxtanx

Question 15

d/dx(cscx)=?

Answer 15

−cscxcotx

Question 16

d/dx(ƒ(g(x)))=?

Answer 16

*We want dy/dx=dƒ/dx=ƒ′. *Let t=g(x), –then ƒ′(x)=dy/dt –and g’(x)=dy/dx *dy/dt ⋅ dt/dx=dy/dx *So, d/dx(ƒ(g(x)))=ƒ′(g(x)) ⋅ g’(x) (1). Differentiate the outside (2). Leave the inside alone (3). Multiply by the derivative of the inside

Question 17

d/dx(e^(ƒ(x)))=?

Answer 17

In general, d/dx(e^(ƒ(x)))=ƒ′(x)e^(ƒ(x))

Question 18

d/dx(a^x)=?

Answer 18

ln(a)a^x

Question 19

d/dx(a^(ƒ(x)))=?

Answer 19

ln(a)ƒ′(x)a^(ƒ(x))

Question 20

d/dx(ln(x))=?

Answer 20

1/x

Question 21

d/dx(ln(ƒ(x)))=?

Answer 21

ƒ′(x)/ƒ(x)=1/ƒ(x)⋅ƒ′(x)

Question 22

d/dx(logₐ(x))=?

Answer 22

1/ln(a)⋅1/x

Question 23

Exponential change-of-base:

Answer 23

base=e^ln(base)

Question 24

Logarithmic Differentiation:

Answer 24

1. Take natural log of both sides. 2. Apply Rules/Properties of Logs to expand the function. 3. Differentiate both sides. 4. Solve for ƒ′(x).

Question 25

d/dx(arcsin(x))=?

Answer 25

1/√(1−x²)

Question 26

d/dx(arctan(x))=?

Answer 26

1/(1+x²)

Question 27

d/dx(arcsec(x))=?

Answer 27

1/∣x∣√(x²−1)

Question 28

d/dx(arccos(x))=?

Answer 28

−1/√(1−x²)

Question 29

d/dx(arccot(x))=?

Answer 29

−1/(1+x²)

Question 30

d/dx(arccsc(x))=?

Answer 30

−1/∣x∣√(x²−1)

Question 31

d/dx(cu)=?

Answer 31

cu’