📘 calculus, algebra

1. **Problem statement:** (a) Given the function $y = 1 - 3e^{-x^2}$, we need to:

1. **Problem 1: Find the instantaneous rate of change of** $f(x) = 3x^2 - 6x$ **at** $x=1$. The instantaneous rate of change of a function at a point is given by the derivative eva

1. **Problem 1:** Find $f'(2)$ for $f(x) = \sqrt{6 - x}$. - The function can be rewritten as $f(x) = (6 - x)^{\frac{1}{2}}$.

1. **Problem 9(a): Find $\frac{dy}{dx}$ when $x=2$ for $y=2x^3 - 6x^2 - 18x + 4$.** 2. The derivative formula for a polynomial $y=f(x)$ is $\frac{dy}{dx} = \frac{d}{dx}(2x^3) - \fr

1. **Problem 1: Find the derivative $\frac{dy}{dx}$ of $f(x) = e^{\sin(x)}$.** 2. The formula for the derivative of an exponential function with a function in the exponent is:

1. **Problem 14:** Given a differentiable function $f$ on $(0,2)$ with derivative $f'(x) = x^2 - x + 1$. We analyze the truth of the statements: 2. **Analyze the derivative:** The