The product and quotient rules Problem set A

Differentiate with respect to x:

1. (a) \(y=x^5\sin x\)
2. (b) \(y=x^{3/2}e^x\)
3. (c) \(y=x^6\ln x\)

Differentiate with respect to x:

1. (a) \(f(x)=x^3\sin4x\)
2. (b) \(f(x)=x^2e^{5x-2}\)
3. (c) \(f(x)=x^3\ln(2x+5)\)

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

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

Differentiate, and where asked evaluate to 3 significant figures.

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

Give exact answers.

1. (a) Find the equation of the tangent to \(y=\dfrac{x}{x^2+1}\) at \(x=2\).
2. (b) Find the equation of the normal to \(y=x\sin3x\) at \(x=\dfrac{\pi}{6}\).

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

1. (a) \(y=e^{2x}-5x\), where the gradient is 3.
2. (b) \(y=x\ln2x\), where the gradient is 3.

The concentration of a drug in the blood, in mg/l, is modelled by \(C=12te^{-0.5t}\), where t is the time in hours after an injection, \(t\ge0\).

1. (a) Find \(\dfrac{dC}{dt}\).
2. (b) Find when the concentration is greatest, and its value then.
3. (c) Find the rate of change of the concentration when \(t=4\), with its units, and say what it tells you.

The concentration of a pollutant in a lake, in mg/l, t days after a spill is \(C=\dfrac{20t}{t^2+16}\), \(t\ge0\).

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

Differentiate and simplify.

1. (a) If \(f(x)=x\sqrt{5-x}\), show that \(f'(x)=\dfrac{ax+b}{2\sqrt{5-x}}\), where a and b are integers to be found.
2. (b) Find and simplify an expression for the derivative of \(f(x)=\dfrac{x}{\sqrt{k-x^2}}\), where k is a positive constant.

Show that \(f(x)=\dfrac{x^2}{x-3}\) 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 \(f(x)=\cot x\), where \(\cot x=\dfrac{\cos x}{\sin x}\), then \(f'(x)=-\dfrac{1}{\sin^2x}\).

Let \(y=\cos^2x\), 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 \(\cos2x=1-2\sin^2x\), differentiate both sides to show that \(\sin2x=2\sin x\cos x\).
4. (d) Given that \(\cos3x=4\cos^3x-3\cos x\), use differentiation to show that \(\sin3x=3\sin x-4\sin^3x\).

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

1. (a) Show that \(\dfrac{dy}{dx}=3y\).
2. (b) Is it true that if \(\dfrac{dy}{dx}=3y\), then \(y=e^{3x}\)? Justify your answer.
3. (c) Now let \(y=e^{kx}\), where \(k>0\). The tangent to this curve at \(x=\dfrac2k\) is L. Prove that the y-intercept of L is independent of k.

Find the x-coordinate of every point on the curve \(y=\ln(x^2-5)\) where the tangent is parallel to the line \(y=4x+3\).