The Chain Rule (with others)
f'(x), if f(x)=sin(x2-4)
What is f'(x)=2xcos(x2-4)?
( f=sin(x2-4) --> cos(x2-4) times (2x) --> f'(x)=2xcos(x2-4) )
dy/dx, if x3y5+3x=8y+1
What is dy/dx=(3x2y5+3)/(8-5x3y4)?
( x3y5+3x=8y+1 --> 3x2y5+5x3y4(dy/dx)+3=8(dy/dx) --> (dy/dx)(8-5x3y4)=3x2y5+3 --> dy/dx=(3x2y5+3)/(8-5x3y4) )
A function whose relation to the variable is given by an equation for which the function has not been solved for a specific variable.
(Hint: Don't overthink it!)
What is an Implicit Function?
If f(x)=cos(3x2+x), f'(x)=(6x+1)sin(3x2+x)
What is False?
f'(x)= -(6x+1)sin(3x2+x)
If 2y=x2+siny, dy/dx=2x/(2-cos(y))
What is True?
( 2y=x2+siny --> 2(dy/dx)=2x+cos(y)(dy/dx) --> (dy/dx)(2-cos(y))=2x --> dy/dx=2x/(2-cos(y)) )
f'(x), if f(x)=sin(cos(x))
dy/dx, if x2tan(y)+sec(x)=2x
What is (dy/dx)=(2-2xtan(y)-sec(x)tan(x))/(x2sec2(y))?
( x2tan(y)+sec(x)=2x --> 2xtan(y)+x2sec(y)+sec(x)tan(x)=2 --> (dy/dx)(x2sec2(y))=2-2xtan(y)-sec(x)tan(x) --> (dy/dx)=(2-2xtan(y)-sec(x)tan(x))/(x2sec2(y)) )
If f=f(g(x)), then f'(x)= f'(g(x))g'(x)
What is the "Outside-inside" Rule?
( Just incase you don't understand the 'If, then' statement --> (In words: Differentiate the "outside" function f and leave "inside" function g(x) alone; then multiply the derivative of the "inside" function.) )
If f(x)=cos3(4x3+2x) = f'(x)(36x2+6)cosx2(4x3+2x)(-sin(4x3+2x))
What is True?
( f(x)=cos3(4x3+2x) --> 3cos2(4x3+2x)(-sin(4x3+2x))(12x2+2) --> f'(x)=(36x2+6)cosx2(4x3+2x)(-sin(4x3+2x)) )
If x2+y2=36, dy/dx=(2x)/(2-cosy)
What is True?
( (d/dx)x2 + (d/dx)y2= (d/dx)36 --> 2x+2y(dy/dx)=0 --> 2y(dy/dx)=-2x --> dy/dx= -x/y --> y=+/-√(36-x2), (dy/dx)=+/- x/√(36-x2) )
h'(x), if h(x)=(x2-3)(√(5x-4))
What is h'(x)=(5(x2-3))/(2√(5x-4))+2x√(5x-4)?
( h(x)=(x2-3)(√(5x-4))--> (x2-3)((5x-4)1/2) --> (x2-3)(1/2)(5x-4)-1/2(5)+(5x-4)1/2(2x) --> h'(x)=(5(x2-3))/(2√(5x-4))+2x√(5x-4) )
y'', if 2x3-3y2=7
What is y''=(2x/y)-(x4/y3)?
( 2x3-3y2=7 --> y'=x2/y --> y''= (2xy-x2y')/y2) --> y''=(2x/y)-(x2/y2)(x2/y) --> y''=(2x/y)-(x4/y3) )
If f○g is the composite of y=f(u) and u=g(x), then:
(f○g)'=f'at u=g(x)*g'at x =f'(g(x))*g'(x)
(Hint: Don't over think it)
What is the Chain Rule?
If f(x)=(7x8+3x2)5/2, f'(x)=(1407+15x)(7x8+3x2)3/2
What is True?
( f(x)=(7x8+3x2)5/2 --> 5/2(7x8+3x2)3/2(56x7+6x) -->f'(x)=(1407+15x)(7x8+3x2)3/2 )
If x3+y3=y+21, then dy/dx=x2/(1-y2)
What is False?
dy/dx=3x2/(1-3y2)
( x3+y3=y+21 --> 3x2+3y2(dy/dx)=(dy/dx) --> 3x2=(dy/dx)(1-3y2) --> dy/dx=3x2/(1-3y2) )
f''(x), f(x)=(7x2-2x+1)3
What is f''(x)=42(7x2-2x+1)2+6(7x2-2x+1)(14x-2)2?
( f(x)=(7x2-2x+1)3 --> f'(x)=3(7x2-2x+1)(14)+(3)(14x-2)(2)(7x2-2x+1)(14x-2) --> f''(x)=42(7x2-2x+1)2+6(7x2-2x+1)(14x-2)2 )
y', at point (2,1), if xy+x2y2=5
What is y'=-1/2?
(xy+x2y2=5 --> xy'+y+2xy2+2x2yy'=0 --> 2y'+1+4+8y'=0 --> y'=-1/2 )
If f is a differentiable function of u and if u is a differentiable function of z, then:
d/dx f(u)= f'(u) du/dx
What is the Power Chain Rule?
If f(x)=(1+sec32x)10 , f'(x)=60(1+sec3(2x))9(tan(2x))
What is False?
f'(x)=60(sec3(2x))(1+sec3(2x))9(tan(2x))?
( f(x)=(1+sec32x)10 --> 10(1+sec3(2x))9(3sec2(2x))(sec(2x)tan(2x))(2) --> f'(x)=60(sec3(2x))(1+sec3(2x))9(tan(2x)) )
If 2x3-3y2=8, y''=(2x/y)-(x4/y3) when y doesn't equal 0.
What is True?
( 2x3-3y2=8 --> y'=(x2/y), when y doens't equal 0 --> y''=(2x/y)-(x2/y2)(y') --> y''=(2x/y)-(x4/y3) )
f'(x), if f(x)=-tan(πx-2)2
What is f'(x)=-6π(πx-2)sec2(πx-2)2
(f(x)=-tan(πx-2)2 --> -3sec2(πx-2)2[2(πx-2)]π --> f'(x)=-6π(πx-2)sec2(πx-2)2 )
y'', if x4+y4=16
What is y''=-48x2/y7?
( x4+y4=16 --> 4x3+4y3y'=0 --> y'=-x3/y3 --> y''=-(3x2y3-3x3y2y')/y6 --> y''=(3x2y3-3x3y2(-x3/y3))/y6 --> y''=(3x2(y4+x4))/y7 --> y''=-3x2(16)/y7 --> y''=-48x2/y7 )
If g(x)=f(x2+3(x-1)+5) and f'(6)=21, then g'(1)=136
What is False?
g'(1)=105
( g'(x)=f'(x2+3(x-1)+5)(2x+3) --> g'(1)=f'(12+3(1-1)+5)(2(1)+3) --> f'(6)(5) --> (21)(5)=105 )
if x2ey+yex=4, at (2,0) y'=8/e2
What is False?
y'=-4/(4+e2)
( x2ey+yex=4 --> x2eyy'+2xey+yex+exy'=0 --> 4y'+4+0+e2y'=0 -->
y'=-4/(4+e2) )