Chain Rule ??? :S?

Question

use the chain rule to differentiate the following
1) (4x+1)^3
2) (2x+1)^7
3) (3x-2)^4
4) (5x+4)^6
5) (3x+7)^5

Answer 1

Okay, the chain rule states that if you have an expression u, and you're taking u to an exponent, you bring the exponent out in front, then lower the exponent by 1, and then multiply your result by the derivative of u (which is du/dx).
So, for number 1:
(4x+1)^3

Bring the three down in front and reduce the exponent by 1 to get:
3(4x+1)^2
Then multiply by the derivative of u, which in this case is 4x+1 (The derivative is 4.)
12(4x+1)^2.

Doing the same thing with the others, you get
(2x+1)^7
= 7(2x+1)^6 * du/dx
= 14(2x+1)^6

(3x-2)^4
= 4(3x-2)^3 * du/dx
= 12(3x-2)^3

(5x+4)^6
= 6(5x+4)^5 * du/dx
= 30(5x+4)^5

(3x+7)^5
= 5(3x+7)^4
= 15(3x+7)^4

By the way, I checked all the answers with my TI-89 calculator, which can take derivatives, so they're all correct.
Hope my explanation helped!! :-)

Answer 2

3*d/dx( 4x+1)=3*4=12
7*d/dx( 2x+1)=7*2=14
4*d/dx( 3x-2)=4*3=12
6*d/dx( 5x+4)=6*5=30
5*d/dx( 3x+7)=5*3=15

Answer 3

u(x) = 4x+1 | u'(x) = 4
v(x) = x^3 | v'(x) = 3 * x^2
f(x) = v(u(x))

Chain rule: f'(x) = v'(u(x)) * u'(x)
f'(x) = 3*(4x+1)^2 * 4
f'(x) = 12*(4x+1)^2