The product and quotient rules Problem set B

Differentiate with respect to x:

1. (a) \(y=x^7\cos x\)
2. (b) \(y=x^{5/2}e^x\)
3. (c) \(y=x^8\ln x\)

Differentiate with respect to x:

1. (a) \(f(x)=x^2\cos5x\)
2. (b) \(f(x)=x^3e^{3x+4}\)
3. (c) \(f(x)=x^2\ln(4x+1)\)

Use the quotient rule to find \(\dfrac{dy}{dx}\). Simplify your answers.

1. (a) \(y=\dfrac{4x-3}{x^2+5}\)
2. (b) \(y=\dfrac{x-5}{\sqrt{2x+1}}\)
3. (c) \(y=\dfrac{e^x}{(5x+2)^2}\)
4. (d) \(y=\dfrac{e^x}{(2-x)^4}\)

Differentiate, and where asked evaluate to 3 significant figures.

1. (a) If \(f(x)=xe^{4x}\), find \(f'(0.5)\).
2. (b) \(y=x^3\sin x\ln x\)
3. (c) \(y=e^{x\sin2x}\)

Give exact answers.

1. (a) Find the equation of the tangent to \(y=\dfrac{x}{x^2+3}\) at \(x=1\).
2. (b) Find the equation of the normal to \(y=x\cos2x\) at \(x=\dfrac{\pi}{4}\).

Find the coordinates of the point where the gradient of the curve is as given. Give coordinates to 3 significant figures.

1. (a) \(y=e^{3x}-4x\), where the gradient is 5.
2. (b) \(y=x\ln3x\), where the gradient is 4.

The profit, in thousands of pounds, from a new app is modelled by \(P=8te^{-0.25t}\), where t is the time in months after the launch, \(t\ge0\).

1. (a) Find \(\dfrac{dP}{dt}\).
2. (b) Find when the profit is greatest, and its value then.
3. (c) Find the rate of change of the profit when \(t=8\), with its units, and say what it tells you.

The number of visitors, in thousands, on a website t days after it opens is \(V=\dfrac{30t}{t^2+25}\), \(t\ge0\).

1. (a) Find \(\dfrac{dV}{dt}\), and simplify.
2. (b) Find the rate of change of the number of visitors when \(t=3\), with its units.
3. (c) Find when the number of visitors is greatest, and its value then.

Differentiate and simplify.

1. (a) If \(f(x)=x\sqrt{1+3x}\), show that \(f'(x)=\dfrac{ax+b}{2\sqrt{1+3x}}\), where a and b are integers to be found.
2. (b) Find and simplify an expression for the derivative of \(f(x)=\dfrac{x}{\sqrt{k+2x^2}}\), where k is a positive constant.

Show that \(f(x)=\dfrac{x^2}{x-4}\) is an increasing function for \(x>a\) and for \(x<b\), where a and b are constants to be found.

Use the quotient rule to show that if \(f(x)=\sec x\), where \(\sec x=\dfrac{1}{\cos x}\), then \(f'(x)=\dfrac{\sin x}{\cos^2x}\).

Let \(y=2\sin^2x\), for \(0<x<2\pi\).

1. (a) Find \(\dfrac{dy}{dx}\).
2. (b) Find the values of x where the gradient of the curve is zero.
3. (c) Given that \(\cos2x=2\cos^2x-1\), differentiate both sides to show that \(\sin2x=2\sin x\cos x\).
4. (d) Given that \(\sin4x=2\sin2x\cos2x\), use differentiation to show that \(\cos4x=\cos^22x-\sin^22x\).

Let \(y=e^{4x}\).

1. (a) Show that \(\dfrac{dy}{dx}=4y\).
2. (b) Is it true that if \(\dfrac{dy}{dx}=4y\), then \(y=e^{4x}\)? Justify your answer.
3. (c) Now let \(y=e^{kx}\), where \(k>0\). The tangent to this curve at \(x=\dfrac3k\) is L. Prove that the y-intercept of L is independent of k.

Find the x-coordinate of every point on the curve \(y=\ln(x^2-6)\) where the tangent is parallel to the line \(y=3x-1\).