Question

Can someone help me with basic derivatives: 3x^2 + 5x + 5?

Answer 1

6x + 5

Answer 2

3(2X) + 5

Answer 3

6x+5

Answer 4

The rules are:
derivative of x^n = n * x^(n-1)
derivative of sum is the sum of the derivatives.
derivative of a constant times a function equals the constant times the derivative of the function.

Answer 5

The approach is pretty straightforward -- \(x^2\) on its own derives to 2x. Since it has a 3 in front of it, we get 3(2x), as above.

x derives to \(x^0\), which is the same as 1. Since it has a 5 in front of it, we get 5, as above.

5 disappears, because it is a constant and the derivative of a constant is 0.

Answer 6

So d/dx (3x^2 +5x + 5) =

d/dx (3x^2) + d/dx (5x) + d/dx (5) =
3* d/dx (x^2) + 5 * d/dx (x) + 0 =
3 * 2 x + 5 * 1 =
6x + 5

Answer 7

\[\lim_{h\rightarrow 0}\ \frac{f(x+h)-f(x)}{h}\]
\[(3(x+h)^2 +5(x+h) +5)-(3x^2 +5x +5)//h\]
\[(3(x^2+2xh+h^2) +5x+5h +5 -3x^2 -5x -5)//h\]
\[3x^2+6xh+3h^2 +5x+5h +5 -3x^2 -5x -5)//h\]
\[\frac{6xh+3h^2+5h}{h}\]
\[\lim_{h\rightarrow0}\ 6x+3h+5\ =\ 6x+5\]

Answer 8

the long version lets you appreciate the shortcut rules ;)