Differentiation

(a) Suppose the following facts are known about the function $g$ and its derivative: $g(5)=1$, $g'(5)=-3$. If $f(x)=\tan(e^{g(x)})$, find the value of $f'(5)$.

(i) Show that $\dfrac{d}{dx}\left(\cos^{-1}\dfrac{2x}{1+x^2}\right)=\dfrac{-2}{1+x^2}$ for $-1\le x\le1$. Explain, in your working, how you use the condition $-1\le x\le1$ to conclude your answer.

The diagram shows the graph of $y=f(x)$. The curve passes through the origin $O$, the points $A(-4a,0)$ and $C(-2a,0)$. The points $B(-3a,2)$ and $D(-a,-2)$ are stationary points and the lines $y=0$ and $x=a$ are asymptotes of the curve.

The diagram below shows the graph of $y=f(x)$. The curve cuts the axes at $(-1,0)$ and $(0,3)$. It has a maximum point at $(3,5)$, and asymptotes $x=-2$ and $y=2$.

The diagram above shows the graph of the function $y=f(x)$ where $f(x)=\dfrac{ax^2+bx+c+1}{x+d}$, $a,b,c,d\in\mathbb R$. The oblique asymptote intersects the axes at $x=\dfrac32$ and $y=3$. Find the values of $a,b,d$.

The diagram below shows the graph of $y=f'(x)$. The curve passes through the origin and has turning points at $(-3,0)$ and $(-1.5,-1)$. The $x$-axis and $x=1$ are the two asymptotes of the curve.

The graphs of $y=f'(x)$ and $y=|f(x)|$ are shown below. The gradient graph has turning point $(-1,-2)$, intercepts $O$ and $(3,0)$, and asymptotes $y=0$, $x=2$. The absolute-value graph shows $(0,1)$, $(1,0)$, $(3,3)$ and asymptotes $y=0$, $x=2$.