The graph of function $f$ contains the points $P(1, 2)$ and $Q(s, r)$. The equation of the secant line through $P$ and $Q$ is $y = \left( \frac{s^2 + 2s - 3}{s - 1} \right) x - 1 - s$. The value of $f'(1)$ is:

If the slope of the tangent drawn at any point $(x, y)$ on the curve $y=f(x)$ is $(6x^2+10x-9)$ and $f(2)=0$, then $f(-2)=$

The differential of ${e^{x^3}}$ with respect to $\log_e x$ is

$f(x)$ and $g(x)$ are differentiable functions such that $\frac{f(x)}{g(x)} = c$,where $c$ is a non-zero constant. If $\frac{f^{\prime}(x)}{g^{\prime}(x)} = \alpha(x)$ and $\left(\frac{f(x)}{g(x)}\right)^{\prime} = \beta(x)$,then $\frac{\alpha(x) - \beta(x)}{\alpha(x) + \beta(x)} = $

For some constants $a$ and $b$,find the derivative of $\frac{x-a}{x-b}$.

If $g(x) = (x^2 + 2x + 1) \cdot f(x)$ such that $f(0) = 5$ and $\lim_{x \to 0} \frac{f(x) - 5}{x} = 4$, then $g'(0) = $