What is the second derivative of \(f\) in terms of \(f'\)?

For position \(s(t)\), what physical quantities do \(s'(t)\) and \(s''(t)\) represent?

Find \(f''(x)\) for \(f(x)=x^4\).

Find \(d^2y/dx^2\) for \(y=7x^2-3x+1\).

Find the first three derivatives of \(f(x)=x^5\).

For \(s(t)=3t^3-2t\), find the acceleration \(a(t)\).

Find \(f''(x)\) for \(f(x)=\sin x+\cos x\).

Find \(y'''\) for \(y=2x^4-x^2\).

Find \(f''(x)\) for \(f(x)=e^{2x}\).

Find \(d^2y/dx^2\) for \(y=\ln x\), with its domain condition.

A particle has \(s(t)=t^4-4t^2\). Find \(v(t)\), \(a(t)\), and \(a(1)\).

For \(f(x)=x^3-6x^2+9x\), use \(f''(x)\) to test the stationary point at \(x=1\).

Find \(f^{(4)}(x)\) for \(f(x)=x^6\).

Find the pattern for the derivatives of \(\sin x\) up to the fourth derivative.

For \(f(x)=x^n\), where \(n\) is a positive integer, find \(f''(x)\).

Find the acceleration for \(s(t)=A\cos(\omega t)\), where \(A\) and \(\omega\) are constants.

Find \(f''(x)\) for \(f(x)=(x^2+1)^3\).

A student says \(f''(x)\) is always positive because it is a derivative of a derivative. Give a counterexample.

Find all \(x\) where \(f(x)=x^4-4x^3\) has zero second derivative, and interpret them as possible inflection points.

For \(f(x)=e^{kx}\), show that the second derivative is proportional to \(f(x)\).