Question

Question one a) Find the derivative of $ f(x)=\sqrt{2 x^{3}+6} $ b) Find $ \frac{d y}{d x} $ when $ x=-4 \cos \theta $ and $ y=3 \sin \theta $ c) Find the derivative of $ f(x)=\frac{\sin x}{1+\cos x} $ d) Find the equation of a tangent line at the given points if $ f(x)=10 x-2 x^{2} \quad $ at $ P(3,12) $ Question two a) Find the derivative of $ f(x)=\frac{2 x^{2}+7}{2 x+1} $ b) Let $ f(x)=\left\{\begin{array}{rr}x-3, & x \leq-1 \\ x^{2}+1, & x>-1\end{array}\right. $ Find (i) $ \lim _{x \rightarrow-1^{-}} f(x) $ (ii) $ \lim _{x \rightarrow-1^{+}} f(x) $ (iii) $ \lim _{x \rightarrow-1} f(x) $ c) Differentiate with respect to $ \mathrm{x}: \mathrm{y}=\log _{8} x^{4} $ d) Find $ \frac{d y}{d x} $ if $ y=\sin x \tan x $

          Question one
a) Find the derivative of \( f(x)=\sqrt{2 x^{3}+6} \)
b) Find \( \frac{d y}{d x} \) when \( x=-4 \cos \theta \) and \( y=3 \sin \theta \)
c) Find the derivative of \( f(x)=\frac{\sin x}{1+\cos x} \)
d) Find the equation of a tangent line at the given points
if \( f(x)=10 x-2 x^{2} \quad \) at \( P(3,12) \)
Question two
a) Find the derivative of \( f(x)=\frac{2 x^{2}+7}{2 x+1} \)
b) Let \( f(x)=\left\{\begin{array}{rr}x-3, & x \leq-1 \\ x^{2}+1, & x>-1\end{array}\right. \)

Find
(i) \( \lim _{x \rightarrow-1^{-}} f(x) \)
(ii) \( \lim _{x \rightarrow-1^{+}} f(x) \)
(iii) \( \lim _{x \rightarrow-1} f(x) \)
c) Differentiate with respect to \( \mathrm{x}: \mathrm{y}=\log _{8} x^{4} \)
d) Find \( \frac{d y}{d x} \) if \( y=\sin x \tan x \)

$Question one a) Find the derivative of f(x)=√(2 x^3+6) b) Find (d y)/(d x) when x=-4 cosθ and y=3 sinθ c) Find the derivative of f(x)=(sin x)/(1+cos x) d) Find the equation of a tangent line at the given points if f(x)=10 x-2 x^2 at P(3,12) Question two a) Find the derivative of f(x)=(2 x^2+7)/(2 x+1) b) Let f(x)={ x-3, x ≤-1 x^2+1, x>-1 . Find (i) limx →-1^- f(x) (ii) limx →-1^+ f(x) (iii) limx →-1 f(x) c) Differentiate with respect to x: y=log8 x^4 d) Find (d y)/(d x) if y=sin x tan x$

Added by Nina P.

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