Question: Derivative of f(x)=ax^n
If f(x) = ax^n, n \in N, a \in R,
then f'(x) = anx^{n-1}.
Differentiate each of the following.
a. y = x^8
b. y = 3x^2
c. y = 7
d. y = 5x^3 - 6x^2
1. **State the problem:** Differentiate the functions given using the power rule for derivatives.
2. **Recall the power rule:** If $$f(x) = ax^n$$ where $$n \in \mathbb{N}$$ and $$a \in \mathbb{R}$$, then the derivative is $$f'(x) = an x^{n-1}$$.
3. **Differentiate each function:**
a. $$y = x^8$$
Using the rule, $$a=1$$ and $$n=8$$.
$$y' = 1 \times 8 x^{8-1} = 8x^7$$
b. $$y = 3x^2$$
Here, $$a=3$$ and $$n=2$$.
$$y' = 3 \times 2 x^{2-1} = 6x$$
c. $$y = 7$$
This is a constant function, so its derivative is zero.
$$y' = 0$$
d. $$y = 5x^3 - 6x^2$$
Differentiate each term separately:
For $$5x^3$$, $$a=5$$, $$n=3$$, so derivative is $$5 \times 3 x^{3-1} = 15x^2$$.
For $$-6x^2$$, $$a=-6$$, $$n=2$$, so derivative is $$-6 \times 2 x^{2-1} = -12x$$.
Combine:
$$y' = 15x^2 - 12x$$
4. **Final answers:**
a. $$y' = 8x^7$$
b. $$y' = 6x$$
c. $$y' = 0$$
d. $$y' = 15x^2 - 12x$$