Basic Derivatives
y = 3x2
y' = 6x
y = 3x2 sin(x)
y' = 3x2 cos(x) + 6x sin(x)
Find the 2nd derivative of y = ex
y'' = ex
Find the equation of the tangent line to y = ln(x3) at x=e.
y-3=3/e (x-e) or just y = 3x/e
The points at which a function is not differentiable.
What is a sharp corner, a cusp, and a vertical tangent line?
f(x) = 5
f'(x) = 0
f(x) = (3x - 2x2)(5 + 4x)
-24x2 + 4x + 15
If the position of an object on the x-axis is given by x(t) = t*cos(3t) meters at time t seconds, then what is the velocity of the object when t = pi/2?
3pi/2 meters per second
What is the definition of a derivative?
What is the limit as h approaches 0 of f(x+h)-f(x) divided by h.
f(x) = 6x3 + x - 2
f'(x) = 18x2 + 1
y = (5x - 2)/(x2 + 1)
y' = (-5x2 + 4x + 5)/(x2 + 1)2
What is the rate of change of f(t) = t2ln(2t-3) at the moment that t = 2?
8 units of f per unit of t
If f is differentiable at x=c, then f is continuous at x=c. True or false?
True
f(x) = 3sin(x) - 2cos(x)
3cos(x) + 2sin(x)
h(x) = (2x + 5) / x1/2
h'(x) = (2x - 5)/(2x3/2)
f(x) = ln (x/(x2 + 1))
f'(x) = (1/x) - (2x/(x2 + 1))
What value of x does f(x)=xe-2x have a horizontal tangent line?
What is x = 1/2 ?
The equation of the tangent line of f(x)=x^2 at x=2
What is y-4=4(x-2)
s(t) = 13t4 + 3t2 - 6t +3
s'(t) = 52t3 + 6t - 6
Blue(x) = 3x sin(3x)
Blue'(x) = 3sin(3x) + 9x cos(3x)
y = ln(5x2+1)7
70x/(5x2+1)
For what values of k does the tangent line to y = ln(x) at x=k have a y-intercept of (0,1)?
k = e2
Find the equation of the normal line of f(x)=(x^2+2x)(x+1) at x=1