The chain rule Problem set A

Find \(\frac{dy}{dx}\) for each.

1. (a) \(y=\left(4 x + 1\right)^{5}\)
2. (b) \(y=\left(x^{3} - 2 x\right)^{4}\)

Find \(\frac{dy}{dx}\) for each.

1. (a) \(y=\left(x^{2} + 5\right)^{\frac{2}{3}}\)
2. (b) \(y=\sqrt{3 x^{2} - x}\)

Find \(\frac{dy}{dx}\) for each.

1. (a) \(y=\sin{\left(5 x \right)}\)
2. (b) \(y=\cos{\left(3 x - 2 \right)}\)

Find \(\frac{dy}{dx}\) for each.

1. (a) \(y=e^{5 x}\)
2. (b) \(y=e^{x^{2} - 4 x}\)

Find \(\frac{dy}{dx}\) for each.

1. (a) \(y=\ln{\left(3 x \right)}\)
2. (b) \(y=\ln{\left(x^{2} + 4 \right)}\)

Find \(\frac{dy}{dx}\) for each. Each needs three derivatives multiplied.

1. (a) \(y=\cos^{2}{\left(2 x \right)}\)
2. (b) \(y=e^{\sin{\left(3 x \right)}}\)
3. (c) \(y=\sqrt{\ln{\left(5 x \right)}}\)

Find \(\frac{dy}{dx}\) for each. Name the inner function first.

1. (a) \(y=\frac{1}{e^{4 x}}\)
2. (b) \(y=\sqrt{\cos{\left(x \right)}}\)
3. (c) \(y=\sin{\left(\sqrt{x} \right)}\)

Find the gradient of the curve \(y=(e^x)^3+2x\) at the point where \(x=\ln 2\).

The temperature of a drink, in °C, t minutes after it is poured is modelled by \(T=18+70e^{-0.05t}\). Find the rate of change of the temperature when \(t=10\). Give the units.

The height of the tide in a harbour, in metres, t hours after midnight is modelled by \(h=5+2\sin(0.5t)\), with the calculator in radians. Find the rate at which the height is changing at 02:00.

Find the equation of the tangent to the curve \(y=\sqrt{x^2+16}\) at the point where \(x=3\). Write it as \(y=mx+c\).

Find the coordinates of the point on the curve \(y=\ln(3x-4)\) where the tangent is parallel to the line \(y=2x-1\).

The line \(y=-4x+1\) is perpendicular to a tangent of the curve \(y=\ln(x+3)\). Find the equation of that tangent. Give \(y=mx+c\) with c to 3 significant figures.

Find the equation of the tangent to the curve \(y=\cos\left(\frac{2}{x}\right)\) at the point where \(x=\frac{4}{\pi}\). Give \(y=mx+c\) with m and c to 3 significant figures.