The chain rule Problem set B

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

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

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

1. (a) \(y=\left(x^{2} + 7\right)^{\frac{3}{4}}\)
2. (b) \(y=\sqrt{2 x^{2} + 5 x}\)

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

1. (a) \(y=\cos{\left(4 x \right)}\)
2. (b) \(y=\sin{\left(6 x - 1 \right)}\)

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

1. (a) \(y=e^{- 3 x}\)
2. (b) \(y=e^{2 x^{2} + x}\)

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

1. (a) \(y=\ln{\left(7 x \right)}\)
2. (b) \(y=\ln{\left(x^{2} + 9 \right)}\)

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

1. (a) \(y=\sin^{2}{\left(4 x \right)}\)
2. (b) \(y=e^{\cos{\left(5 x \right)}}\)
3. (c) \(y=\sqrt[3]{\ln{\left(2 x \right)}}\)

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

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

Find the gradient of the curve \(y=(e^x)^4+3x\) at the point where \(x=\ln 2\).

A phone battery charges so that its charge level, in per cent, after t hours is modelled by \(C=100-80e^{-0.2t}\). Find the rate at which the charge level is changing when \(t=5\). Give the units.

The height above the ground of a seat on a Ferris wheel, in metres, t minutes after it starts is modelled by \(h=10+6\cos(0.3t)\), with the calculator in radians. Find the rate at which the height is changing when \(t=4\).

Find the equation of the tangent to the curve \(y=\sqrt{x^2+24}\) at the point where \(x=5\). Write it as \(y=mx+c\).

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

The line \(y=-5x+3\) is perpendicular to a tangent of the curve \(y=\ln(x-2)\). 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 \(y=\cos\left(\frac{3}{x}\right)\) at the point where \(x=\frac{6}{\pi}\). Give \(y=mx+c\) with m and c to 3 significant figures.