Differentiate \(x^2\mathrm{e}^{3x}\) with respect to \(x\).

The tangent to the curve \(y=\mathrm{e}^{x}(2 x+5)^{\frac{1}{2}}\) at the point where \(x=2\) meets the \(x\)-axis at the point \(X\) and the \(y\)-axis at the point \(Y\). Find the coordinates of the mid-point of \(X Y\), giving your answer in exact form.

A curve has equation \(y=5 \mathrm{e}^{2 x-1}+\mathrm{e}\). The tangent to the curve at the point where \(x=1\) cuts the \(x\)-axis at the point \(P\).

Find the equation of the tangent in the form \(y=m x+c\), where \(m\) and \(c\) are exact values, and hence find the \(x\)-coordinate of \(P\).

A curve has equation

\(y=x\mathrm e^{2x}\).

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

(b) Find the equation of the normal to the curve at the point where \(x=1\).

(c) Use your answer to part (a) to find the exact value of

\(\displaystyle \int_0^2 2x\mathrm e^{2x}\,\mathrm dx.\)

The curve

\(y=5e^{2x}-3\)

meets the \(y\)-axis at the point \(A\). The tangent to the curve at \(A\) meets the \(x\)-axis at the point \(B\). Find the length \(AB\).

(a) Given that

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

find \(\dfrac{dy}{dx}\).

(b) Hence, given that \(y\) is increasing at the rate of \(2\) units per second, find the exact rate of change of \(x\) when \(x=2\).