A curve has equation \(y=\left(\frac{x^2-1}{x^2+1}\right)^4\).

(a) Show that \(\frac{\mathrm{d}y}{\mathrm{d}x}\) can be written as \(\frac{Ax(x^2-1)^3}{(x^2+1)^5}\), where \(A\) is a positive integer to be found.

(b)(i) Show that the curve has stationary points where \(x=-1\), \(x=0\) and \(x=1\).

(ii) Use the first derivative test to determine which two stationary points have the same nature and state whether they are maximum or minimum points.

It is given that \(y=\frac{\ln(3x^2+16)}{x+2}\).

(a) Find \(\frac{\mathrm{d}y}{\mathrm{d}x}\) when \(x=0\). Give your answer in the form \(\ln p\), where \(p\) is a constant.

(b) Given that \(x\) increases from \(0\) to \(h\), where \(h\) is small, write down the approximate change in \(y\).

Given that \(y=\frac{4 x^{3}-5}{x^{2}}\), show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) can be written as \(\frac{2\left(2 x^{3}+5\right)}{x^{3}}\).

Given that

\(\displaystyle y=\frac{(3x-4)^{\frac13}}{2x+1},\)

show that \(\frac{\mathrm dy}{\mathrm dx}\) can be written in the form

\(\displaystyle \frac{Ax+B}{(2x+1)^2(3x-4)^{\frac23}},\)

where \(A\) and \(B\) are integers. Hence find the coordinates of the stationary point on the curve.

A curve has equation

\(y=\frac{\sqrt{5x-2}}{x-3}\).

(a) Explain why the curve does not exist when \(x\lt \frac25\).

(b) Show that \(\frac{dy}{dx}\) can be written in the form

\(-\frac{Ax+B}{2(x-3)^2\sqrt{5x-2}}\),

where \(A\) and \(B\) are positive integers.

Given that

\(\displaystyle y=\frac{(5x+2)^{\frac13}}{(x-1)^2},\)

show that \(\frac{\mathrm dy}{\mathrm dx}\) can be written in the form

\(\displaystyle -\frac{Ax+B}{3(5x+2)^{\frac23}(x-1)^3},\)

where \(A\) and \(B\) are integers.

Variables \(x\) and \(y\) are such that \(\displaystyle y=\frac{\ln(2x^2-3)}{3x}\).

(a) Find \(\displaystyle \frac{dy}{dx}\).

(b) Hence find the approximate change in \(y\) when \(x\) increases from 2 to \(2+h\), where \(h\) is small.

(c) At the instant when \(x=2\), \(y\) is increasing at the rate of 4 units per second. Find the corresponding rate of increase in \(x\).

Differentiate \(\displaystyle y=\frac{\mathrm{e}^{4x}\tan x}{\ln x}\) with respect to \(x\).

A curve has equation

\(y=\frac{(2x^2+10)^{\frac32}}{x-1}\quad\text{for }x\gt 1.\)

(a) Show that \(\dfrac{dy}{dx}\) can be written in the form

\(\frac{(2x^2+10)^{\frac12}}{(x-1)^2}(Ax^2+Bx+C),\)

where \(A\), \(B\) and \(C\) are integers.

(b) Show that, for \(x\gt 1\), the curve has exactly one stationary point. Find the value of \(x\) at this stationary point.

A curve has equation

\(y=\frac{(x^2-5)^{\frac13}}{x+1}\)

for \(x\gt -1\).

(a) Show that

\(\frac{dy}{dx}=\frac{Ax^2+Bx+C}{3(x+1)^2(x^2-5)^{\frac23}},\)

where \(A\), \(B\) and \(C\) are integers.

(b) Find the \(x\)-coordinate of the stationary point on the curve.

(c) Explain how you could determine the nature of this stationary point. You are not required to find the nature of this stationary point.

It is given that \(y=\dfrac{\ln(2x^3+5)}{x-1}\) for \(x\gt 1\).

(i) Find \(\frac{dy}{dx}\) when \(x=2\).

(ii) Find the approximate change in \(y\) as \(x\) increases from \(2\) to \(2+p\), where \(p\) is small.

Two variables \(x\) and \(y\) are such that \(y=\dfrac{\ln x}{x^3}\), for \(x\gt 0\).

(i) Show that \(\dfrac{dy}{dx}=\dfrac{1-3\ln x}{x^4}\).

(ii) Hence find the approximate change in \(y\) as \(x\) increases from \(e\) to \(e+h\), where \(h\) is small.

Find \(\dfrac{dy}{dx}\) when \(y=\dfrac{\sin x}{\ln x^2}\).

Differentiate with respect to \(x\)

(i) \(4x\tan x\),

(ii) \(\dfrac{e^{3x+1}}{x^2-1}\).

Differentiate with respect to \(x\)

(i) \(4x\tan x\),

(ii) \(\dfrac{e^{3x+1}}{x^2-1}\).

A curve has equation

\(y=\frac{\ln(2x^2+3)}{5x+2}.\)

(i) Show that \(\dfrac{dy}{dx}=-\dfrac54\ln3\) when \(x=0\).

(ii) Hence find the equation of the tangent to the curve at the point where \(x=0\).

It is given that

\(y=\frac{(5x^2+4)^{1/2}}{x+1}.\)

Find the exact value of \(\dfrac{dy}{dx}\) when \(x=3\).

Differentiate with respect to \(x\),

(i) \((1+4x)^{10}\cos x\),

(ii) \(\dfrac{e^{4x-5}}{\tan x}\).