1.07j Differentiate exponentials: e^(kx) and a^(kx)

A curve has equation \(y = x^4 + 2^x\) Find an expression for \(\frac{dy}{dx}\) [2 marks]

A curve has equation \(y = e^{3x}\).

1. Determine the value of \(x\) when \(y = 10\). [2]
2. Determine the gradient of the tangent to the curve at the point where \(x = 2\). [2]

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]

A scientist is studying the growth of two different populations of bacteria. The number of bacteria, \(N\), in the first population is modelled by the equation $$N = Ae^{kt} \quad t \geq 0$$ where \(A\) and \(k\) are positive constants and \(t\) is the time in hours from the start of the study. Given that • there were 1000 bacteria in this population at the start of the study • it took exactly 5 hours from the start of the study for this population to double

1. find a complete equation for the model. [4]
2. Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures. [2]

The number of bacteria, \(M\), in the second population is modelled by the equation $$M = 500e^{1.4t} \quad t \geq 0$$ where \(k\) has the value found in part (a) and \(t\) is the time in hours from the start of the study. Given that \(T\) hours after the start of the study, the number of bacteria in the two different populations was the same,

1. find the value of \(T\). [3]

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]

Curve C has equation $$y = x^3 - 7x^2 + 5x + 4$$ The point \(P(2, -6)\) lies on \(C\) Find the equation of the tangent to \(C\) at \(P\) Give your answer in the form \(y = mx + c\) where \(m\) and \(c\) are integers to be found. [3]

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]