Chain Rule (3.6)

Derivatives

100

y=(x)^3, find dy/dx

dy/dx=3(x)^2

200

y=(x)^4+3(x)^2, find dy/dx

dy/dx=4(x)^3+6x

300

y=(9(x)^2 + 4)^1/2, find dy/dx

dy/dx=9x/(9(x)^2 + 4)^1/2

400

y= (13(x)^2 - 5x + 8)^1/2, find dy/dx

dy/dx=(26x - 5)/ 2(13(x)^2 - 5x +8)^1/2

500

y= 3/(2x + 1)^1/2, find dy/dx

dy/dx= -3(2x + 1)^-3/2

Relating Derivatives

100

(dy/du) x (du/dx) =

dy/dx

200

Let f(x)=sinx, g(x)= x^2 + 1, h(x)=7x, find f(g(x))

sin(x^2 +1)

300

f(x)=cosx, g(x)=(x+2)^1/2, find (cosx+2)^1/2 in these terms

g(f(x))

400

y=(6(x)^2 + 1)^3, find dy/dx

dy/dx= 432(x)^5 + 432(x)^3 + 36x

500

y=(4x + (x)^(-5))^1/3, find dy/dx

dy/dx= (4(x)^6-5) / (3(x)^(8/3) x (4(x)^6 + 1)^2/3

Trigonometry

100

y= sin(3x+1), find dy/dx

dy/dx= 3cos(3x+1)

200

y= tan(x)^1/2, find dy/dx

dy/dx= ((sec^2)(x)^1/2) / 2(x)^1/2

300

y= 5sin((x)^2 + 1), find dy/dx

dy/dx= 10xcos((x)^2 +1)

400

y= (2)^cos(x), find dy/dx

dy/dx= -ln(2)2^cos(x)sin(x)

500

y= (4/3pi)sin3t + (4/5pi)cos5t, find dy/dx

dy/dx= 4/pi (cos3t) - 4/pi (sin5t)

Miscellaneous

100

y= 4(x^3 + 5)^2, find dy/dx

dy/dx= 24(x)^5 + 120(x)^2

200

y= log10e^x, find dy/dx

dy/dx= 1/ln10

300

y= (1 + (cos)^2 (7x))^3, find dy/dx

dy/dx= -42(1 + (cos)^2 (7x))^2 cos7xsin7x

400

y= ln(3(x)^2 + 9x + 4), find dy/dx

dy/dx= (6x+9) / (3(x)^2 + 9x + 4)

500

y= e^(w^4 - 3w^2 + 9), find dy/dx

dy/dx= (4(w)^3 - 6w)(e^(w^4 - 3w^2 + 9))