1.07q Product and quotient rules: differentiation

Differentiate the following with respect to \(x\), simplifying your answers fully

1. \(y = e^{3x} + \ln 2x\) [1]
2. \(y = (5 + x^2)^{\frac{3}{2}}\) [2]
3. \(y = \frac{2x}{(5-3x^2)^{\frac{1}{2}}}\) [4]
4. \(y = e^{-\frac{3}{x}} \ln(1 + x^3)\) [3]

\includegraphics{figure_5} Figure 5 shows a sketch of the curve with equation \(y = f(x)\), where $$f(x) = \frac{4\sin 2x}{e^{\sqrt{2}x-1}}, \quad 0 \leq x \leq \pi$$ The curve has a maximum turning point at \(P\) and a minimum turning point at \(Q\) as shown in Figure 5.

1. Show that the \(x\) coordinates of point \(P\) and point \(Q\) are solutions of the equation $$\tan 2x = \sqrt{2}$$ [4]
2. Using your answer to part (a), find the \(x\)-coordinate of the minimum turning point on the curve with equation $$y = 3 - 2f(x)$$ [2]

\includegraphics{figure_4} Figure 4 shows a sketch of part of the curve with equation $$y = 2e^{2x} - xe^{2x}, \quad x \in \mathbb{R}$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the \(x\)-axis and the \(y\)-axis. Use calculus to show that the exact area of \(R\) can be written in the form \(pe^t + q\), where \(p\) and \(q\) are rational constants to be found. (Solutions based entirely on graphical or numerical methods are not acceptable.) [5]

In this question you must show detailed reasoning. \includegraphics{figure_9} The diagram shows the curve \(y = \frac{4\cos 2x}{3 - \sin 2x}\) for \(x > 0\), and the normal to the curve at the point \((\frac{1}{4}\pi, 0)\). Show that the exact area of the shaded region enclosed by the curve, the normal to the curve and the \(y\)-axis is \(\ln \frac{2}{3} + \frac{1}{128}\pi^2\). [10]

Find the equation of the tangent to the curve $$y = 3x^2(x + 2)^6$$ at the point \((-1, 3)\), giving your answer in the form \(y = mx + c\). [5]

The function \(f\) is defined by $$f(x) = \frac{(x + 5)(x + 1)}{(x + 4)} - \ln(x + 4) \quad x \in \mathbb{R} \quad x > k$$

1. State the smallest possible value of \(k\). [1]
2. Show that $$f'(x) = \frac{ax^2 + bx + c}{(x + 4)^2}$$ where \(a\), \(b\) and \(c\) are integers to be found. [4]
3. Hence show that \(f\) is an increasing function. [2]

Find \(\frac{dy}{dx}\) for the following functions, simplifying your answers as far as possible.

1. \(y = \cos x - 2 \sin 2x\) [2]
2. \(y = \frac{1}{2}x^4 + 2x^4 \ln x\) [3]
3. \(y = \frac{2e^{3x} - 1}{3e^{3x} - 1}\) [3]

In this question you must show detailed algebraic reasoning. Find the coordinates of any stationary points on the curve below. $$y = (1 - 3x)(3 - x)^3$$ [5]

In all questions you must show all stages of your working, justifying solutions and not relying solely on calculator technology.

1. Differentiate with respect to \(x\)
   1. \(x^2 e^{3x + 2}\), [4]
   2. \(\frac{\cos(2x^4)}{3x}\). [4]

In this question you must show detailed reasoning. \includegraphics{figure_5} The function f is defined for the domain \(x \geqslant 0\) by $$\mathrm{f}(x) = (2x^2 - 3x)\mathrm{e}^{-x}.$$ The diagram shows the curve \(y = \mathrm{f}(x)\).

1. Find the range of f. [6]

1. The function g is defined for the domain \(x \geqslant k\) by $$\mathrm{g}(x) = (2x^2 - 3x)\mathrm{e}^{-x}.$$ Given that g is a one-one function, state the least possible value of \(k\). [1]

1. Find the exact area of the shaded region enclosed by the curve and the \(x\)-axis. [7]

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

1. Find \(\frac{dy}{dx}\), giving your answer in the form \(e^{-2x}(ax + b)\), where \(a\) and \(b\) are integers. [3]
2. Determine the set of values of \(x\) for which \(y\) is increasing. [2]