Derivative Rules
what are the derivatives of sinu, secu, and tanu?
cosu * u', secutanu * u', sec2u * u'
If a derivative is positive, the function will be ______.
If a second derivative is positive, the function will be _____.
increasing, concave up.
sin2x+cos2x = ?
1
find y'.
y = 3√ (x2) sinx
3/2 x1/2 sinx + x3/2 cos x
when do we NEED to use logarithmic differentiation?
when there's a variable in the base and exponent
List the 3 types of ways a function fails differentiability. What does this look like in the graph of the derivative?
corner/cusp, discontinuity, vertical tangent line.
jump discontinuities or vertical asymptotes
if you have the equation of the tangent line, how do you find the equation of the normal line?
first find the slope of the normal line, which is the opposite reciprocal of the slope of the tangent line. Then plug into the point slope formula and solve for y.
evaluate: arccos(- √3 / 2)
5pi/6
find h'.
h(x)=sqroot(arctanx)
1 / (2sqroot(arctanx)*(1+x2))
what are the derivatives of arcsinu, arccosu, and arctanu?
u'/root(1-u2), -u'/root(1-u2), u'/(1+u2)
rewrite the following into cartesian:
y = root(t)
x = ln(t)
y = ex/2
what 2 limits = e?
lim as x -> 0 of (1+x)1/x
lim as n -> infinity of (1+1/n)n
find y'.
x2 + y2 = sin(xy)
y' = − ( (ycos(xy) − 2x) / (xcos(xy) − 2y) )
what technique would you use to find the derivative of the following?
x2 + y2 = sin(xy)
implicit differentiation
Given the following graph with 3 functions, identify the function, its derivative, and the second derivative:
A, B, C
find the points on the curve where the tangent line is horizontal.
x = 2t3+ 3t2-12t
y = 2t3+ 3t2+1
(0,1) & (13,2)
lim as x -> 0 of sin9x / sin10x
9/10
find y'
y= 3√(sin(3√(x)))
cos 3√(x) / 9(x sin 3√(x))2/3
OR
1/3(sin 3√(x))-2/3*(cos 3√(x))1/3*x-2/3
What technique would you use to find y'?
y= 3√(sin(3√(x)))
chain rule
Sketch a graph that satisfies the following conditions:
find an equation of the tangent line to the curve at the point corresponding to the given value of the parameter.
x = cos(theta) + sin(2theta)
y = sin(theta) + cos(2theta)
theta = 0
y = 0.5x + 0.5
lim as x -> negative infinity of arcsin(3x3+9x5 / 3x2 - 81 + 18x5)
pi/6
find y'
y = sin(x3) / sin(x2)
y= cos(x3)3x2 sin(x2)−cos(x2)2x sin(x3) / [sin(x2)]2
OR
xcsc(x2)*(3xcos(x3)-2sin(x3)cot(x2))