Sure, let's go through each function step by step to find their derivatives.
### 1. [tex]\( y = x^2 - 4x + 5 \)[/tex]
To find the derivative of this function, we'll apply standard derivative rules:
- The derivative of [tex]\(x^2\)[/tex] is [tex]\(2x\)[/tex].
- The derivative of [tex]\(-4x\)[/tex] is [tex]\(-4\)[/tex].
- The derivative of a constant [tex]\(5\)[/tex] is [tex]\(0\)[/tex].
Putting it all together, we get:
[tex]\[ y' = 2x - 4 \][/tex]
### 2. [tex]\( y = x^3 + 2x \)[/tex]
Similarly, for this function:
- The derivative of [tex]\(x^3\)[/tex] is [tex]\(3x^2\)[/tex].
- The derivative of [tex]\(2x\)[/tex] is [tex]\(2\)[/tex].
Thus, the derivative is:
[tex]\[ y' = 3x^2 + 2 \][/tex]
### 3. [tex]\( y = \cos x \)[/tex]
For the trigonometric function [tex]\(\cos x\)[/tex]:
- The derivative of [tex]\(\cos x\)[/tex] is [tex]\(-\sin x\)[/tex].
So, the derivative is:
[tex]\[ y' = -\sin x \][/tex]
So, summarizing the derivatives:
1. For [tex]\( y = x^2 - 4x + 5 \)[/tex], [tex]\( y' = 2x - 4 \)[/tex]
2. For [tex]\( y = x^3 + 2x \)[/tex], [tex]\( y' = 3x^2 + 2 \)[/tex]
3. For [tex]\( y = \cos x \)[/tex], [tex]\( y' = -\sin x \)[/tex]
These derivatives match the given results.
