Limit Definition of the Derivative
Basic Derivative Rules
Chain, Product, and Quotient Rule
Implicit Differentiation
Differentiability
100
Use the limit definition of the derivative to determine the derivative of the following function:
f(x)=x^2+3x^
The derivative of is...
f'(x)=2x+3
The setup is...
lim_(h->0)([(x+h)^2+3(x+h)]-[x^2+3x])/h
100
Find the derivative.
f(x)=4x^(-1/2)+3x^7-1/2x^4+0.5
f(x)=-2x^(-3/2)+21x^6-2x^3
100
Find the derivative:
y=sin(2x)
dy/dx=2cos(2x)
100
Find dy/dx.
3x^2+3y^2=2
dy/dx=-x/y
100
B
200
Use the limit definition of the derivative to determine the derivative of the following function:
f(x)=sqrt(x)
The derivative of is...
f'(x)=1/(2sqrt(x)
The setup is...
lim_(h->0)(sqrt(x+h)-sqrt(x))/h
200
Find the derivative:
y=sinx+lnx-tan^-1x
dy/dx=cosx+1/x-1/(1+x^2)
200
Find the derivative:
y=ln(1/2x)
dy/dx=1/(1/2x)*1/2
or...
dy/dx=1/x
200
Find the value of dy/dx at (pi/2,1).
sin(xy)=y
dy/dx=0
200
What is the alternative definition of the derivative we use to determine the derivative at a particular point?
lim_(x->a)(f(x)-f(a))/(x-a)
300
Use the limit definition of the derivative to determine the derivative of the following function:
f(x)=1/(x+2)
The derivative of is...
f'(x)=-1/(x+2)^2
The setup is...
lim_(h->0)(1/((x+h)+2)-(1/(x+2)))/h
300
Find the derivative:
y=ln(e^x)+e^(lnx)
y=2
300
Write the equation of the tangent line for the function
f(x)=(2x+4)^2+3x
at x=1.
y-39=27(x-1)
or
y=27x+12
300
Find dy/dx.
3x^2y^2=4x^2-4xy
dy/dx=(4x-2y-3xy^2)/(3x^2y+2x)
300
Determine the values of a and b that make the following function differentiable on its domain.
f(x)={(ax+b, x> -1),(bx^2-3, x<=-1):}
a=3 and b=-3/2
400
Use the limit definition of the derivative to determine the derivative of the following function:
f(x)=4-sqrt(x+3)
The derivative of is...
f'(x)=-1/(2sqrt(x+3))
The setup is...
lim_(h->0)(4-sqrt(x+h+3)-(4-sqrt(x+3)))/h
400
Find the value of the derivative at x=3:
h(x)=2f(x)+1/3g(x)
17
400
Find the derivative:
f(x)=cos^2(3x)
f'(x)=2cos(3x)*-sin(3x)*3
or
f'(x)=-6cos(3x)sin(3x)
400
Find d2y/dx2.
x^2+y^2=100
(d^2y)/dx^2=-100/y^3
400
B
500
Use the limit definition of the derivative to determine the derivative of the following function:
f(x)=(x+1)/(2-x)
The derivative of is...
f'(x)=-1/(x+2)^2
The setup is...
lim_(h->0)(((x+h)+1)/(2-(x+h))-(x+1)/(2-x))/h
500
Find the derivative:
f(x)=cscx+logx
f'(x)=-cscxcotx+1/(xln10)
500
Find the derivative:
y=tan(cos(sinx))
dy/dx=sec^2(cos(sinx))*-sin(sinx)*cosx
or
dy/dx=-sec^2(cos(sinx))*sin(sinx)*cosx
500
Find dy/dx.
cos(x^2+2y)+xe^(y^2)=1
dy/dx=(2xsin(x^2+2y)-e^(y^2))/(2yxe^(y^2)-2sin(x^2+2y)
500
Prove (formally!) whether the following function is continuous and differentiable (or not!).
f(x)={(x^2+x-7, x>=2),(5x-11, x<2):}
Continuous and differentiable at all x-values.
See work.