Derivative Evaluation 5F2F21

1. **State the problem:** We are given values and derivatives of functions $f$ and $g$ at $x=5$: $$f(5) = -2, \quad g(5) = 8, \quad f'(5) = 1, \quad g'(5) = 4$$ We need to find the derivatives at $x=5$ of two expressions: - $$\frac{d}{dx} \left(3[f(x)]^2 + 5f(x) - 2\right)$$ - $$\frac{d}{dx} \left(\frac{1}{3 - 4g(x)}\right)$$ --- 2. **Formula and rules:** - For the first expression, use the sum rule and chain rule: $$\frac{d}{dx} \left(3[f(x)]^2\right) = 3 \cdot 2 f(x) f'(x) = 6 f(x) f'(x)$$ $$\frac{d}{dx} \left(5 f(x)\right) = 5 f'(x)$$ Derivative of constant $-2$ is $0$. - For the second expression, use the chain rule and derivative of reciprocal function: $$y = \frac{1}{u}, \quad \frac{dy}{dx} = -\frac{1}{u^2} \frac{du}{dx}$$ where $$u = 3 - 4 g(x)$$ and $$\frac{du}{dx} = -4 g'(x)$$ --- 3. **Calculate the first derivative at $x=5$:** $$\frac{d}{dx} \left(3[f(x)]^2 + 5 f(x) - 2\right) = 6 f(x) f'(x) + 5 f'(x)$$ Substitute values: $$= 6 \times (-2) \times 1 + 5 \times 1 = -12 + 5 = -7$$ --- 4. **Calculate the second derivative at $x=5$:** First, $$u = 3 - 4 g(x)$$ At $x=5$: $$u = 3 - 4 \times 8 = 3 - 32 = -29$$ Derivative: $$\frac{du}{dx} = -4 g'(x)$$ At $x=5$: $$\frac{du}{dx} = -4 \times 4 = -16$$ Now, $$\frac{d}{dx} \left(\frac{1}{3 - 4 g(x)}\right) = -\frac{1}{u^2} \frac{du}{dx} = -\frac{1}{(-29)^2} (-16) = \frac{16}{841}$$ --- **Final answers:** 1. $$\frac{d}{dx} \left(3[f(x)]^2 + 5 f(x) - 2\right) \bigg|_{x=5} = -7$$ 2. $$\frac{d}{dx} \left(\frac{1}{3 - 4 g(x)}\right) \bigg|_{x=5} = \frac{16}{841}$$