Determine f'(x) for the following functions: a) f(x) = (x + 5)^2 b) f(x) = (2x - 3)^2 c) f(x) = (x - 1)^3

Steps to Solve

1. Differentiate $f(x) = (x + 5)^2$

Apply the power rule and chain rule. The power rule states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. The chain rule states that if $f(x) = g(h(x))$, then $f'(x) = g'(h(x)) \cdot h'(x)$.

So, $f'(x) = 2(x+5)^{2-1} \cdot \frac{d}{dx}(x+5) = 2(x+5)(1) = 2x + 10$.

$$f'(x) = 2(x+5)$$ $$f'(x) = 2x + 10$$

2. Differentiate $f(x) = (2x - 3)^2$

Again, use the power rule and the chain rule.

$f'(x) = 2(2x-3)^{2-1} \cdot \frac{d}{dx}(2x-3) = 2(2x-3)(2) = 4(2x-3) = 8x - 12$.

$$f'(x) = 2(2x - 3) \cdot 2$$ $$f'(x) = 4(2x - 3)$$ $$f'(x) = 8x - 12$$

3. Differentiate $f(x) = (x - 1)^3$

Apply the power rule and chain rule one more time.

$f'(x) = 3(x-1)^{3-1} \cdot \frac{d}{dx}(x-1) = 3(x-1)^2(1) = 3(x-1)^2$.

$$f'(x) = 3(x-1)^2$$ $$f'(x) = 3(x^2 - 2x + 1)$$ $$f'(x) = 3x^2 - 6x + 3$$