Basic Derivatives
f(x)=(x2+1)(x3-5)-ln(5)
f'(x)=5x4+3x2-10x
cos(x),-sin(x),sec2(x)
h(x)=ln(3x+1)
h'(x)=(1/3x+1)(3)
x3+y3=25, find dy/dx.
dy/dx = -x2/y2
State three indeterminate forms
∞/∞, 0(∞), ∞-∞, 00, 1infinity, ∞0, 0/0
f(a)=e6-log(a)
f'(a)=0-1/[ln(10)a]
f(x)=sin2x+cos2x
f'(x)=0 via pythagorean identity
1 point if brute forced and get f'(x)=2sin(x)cos(x)+2cos(x)sin(-x)
V(x)=(x2+1)x
V'(x)=(x2+1)x[x(1/x2+1)(2x))+ln(x2+1)]
Give three notations for the first derivative
f'(x), Dx, dy/dx
limx->0 (ex-1)/x
1
f(x)=1000000000x1000000000, find f1000000000(x)
1000000000(1000000000!), where (!) is the factorial.
350 points if the factorial is not used (Dr.Duhon's discretion)
f(k)=sin(cos(sin(tan(k))))
f'(k)=cos(cos(sin(tan(k))))(-sin(sin(tan(k))))(cos(tan(k)))(sec2(k))
g(x)=(xex)/(x2-1). Use logarithmic differentiation to find g'(x).
g'(x)=(xex)/(x2-1)[(1/x)+1-(2x/(x2-1))]
Differentiate yln(x)=(6xy)/(12y) with respect to x
(dy/dx)lnx + (1/x)y = 1/2
limx->0+ xlnx
0
Let f(x),g(x),h(x) be functions. Define w(x)=f(2g(x))+eg(x)h(x). Find w'(x).
w'(x)=f'(2g(x))(2g'(x))+(eg(x)h(x))(g'(x)h(x)+g(x)h'(x))
Let f(x)= cos(4x)sin(2x). find the slope of the tangent line at x=π/6.
f'(x)=cos(4x)(2)cos(2x)-sin(4x)(4)sin(2x),
plug in and f'(π/6)=-7/2
h(x)=ln((x3+1)x/ex^3)
h'(x)=x(1/(x3+1))(3x2)+ln(x3+1)-3x2
Find dy/dx, 2x3+4y4=6xy
dy/dx = (6y-6x)/(16y-6x)
limx->0 (1-cosx)/x2
DNE
f(x)=sin(ln(x2+ln(x3)))
f'(x)=cos(ln(x2+ln(x3)))[1/(x2+ln(x3)][2x+3x2/x3]
sin-1(tan(2+ecos(x)))
Will be written on the board
If a sum of $4000 is deposited into an account that pays r% interest, compounded monthly, the balance after 8 years is given by
A=4000(1+r/1200)96
Find dA/dr when r=6.
Find dz/dt(sin2 (z)+u3+(z/u) = 3zu)
2sin(z)cos(z)(dz/dt)+3u2(du/dt)+[u(dz/dt)-z(du/dt)]/u2 = 3z(du/dt) + 3u(dz/dt)
limx->∞ (1+(1/x))x
e