Review of Derivatives Flashcards
The derivative of ƒ @ a:
lim (ƒ(a+h)−ƒ(a))/((a+h)−a) as h->0 or, lim Δƒ(a)/Δa as h->0
Differential operator notation:
d/dx(ƒ(x))=ƒ′(x)
d/dx(c)=?
0, for any constant c
d/dx(ƒ(x)±g(x))=?
d/dx(u±v)=?
d/dx(ƒ(x))±d/dx(g(x))=ƒ′(x)±g’(x) *differentiation of sums is term-by-term!
u’±v’
d/dx(xⁿ)=?
nxⁿ⁻¹, for any real n
d/dx(kƒ(x))=?
kd/dx(ƒ(x))=kƒ′(x), for any real k and differentiable ƒ
d/dx(sinx)=?
cosx
d/dx(cosx)=?
−sinx
d/dx(e^x)=?
e^x
d/dx(ƒ(x)g(x))=?
d/dx(uv)=?
ƒ′(x)g(x)+ƒ(x)g’(x)
u’v+uv’
d/dx(ƒ(x)/g(x))=?
d/dx(u/v)=?
(ƒ′(x)g(x)−ƒ(x)g’(x))/((g(x))^2)
(v’u−uv’
d/dx(tanx)=?
sec^2x
d/dx(cotx)=?
−csc^2x
d/dx(secx)=?
secxtanx
d/dx(cscx)=?
−cscxcotx
d/dx(ƒ(g(x)))=?
*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
d/dx(e^(ƒ(x)))=?
In general, d/dx(e^(ƒ(x)))=ƒ′(x)e^(ƒ(x))
d/dx(a^x)=?
ln(a)a^x
d/dx(a^(ƒ(x)))=?
ln(a)ƒ′(x)a^(ƒ(x))
d/dx(ln(x))=?
1/x
d/dx(ln(ƒ(x)))=?
ƒ′(x)/ƒ(x)=1/ƒ(x)⋅ƒ′(x)
d/dx(logₐ(x))=?
1/ln(a)⋅1/x
Exponential change-of-base:
base=e^ln(base)
Logarithmic Differentiation:
- 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).
d/dx(arcsin(x))=?
1/√(1−x²)
d/dx(arctan(x))=?
1/(1+x²)
d/dx(arcsec(x))=?
1/∣x∣√(x²−1)
d/dx(arccos(x))=?
−1/√(1−x²)
d/dx(arccot(x))=?
−1/(1+x²)
d/dx(arccsc(x))=?
−1/∣x∣√(x²−1)
d/dx(cu)=?
cu’
