State the derivative formula for an inverse function \(f^{-1}\).

If \(y=f^{-1}(x)\), what equation relates \(x\), \(y\), and \(f\)?

Use inverse differentiation to find \(d(\arcsin x)/dx\).

Use inverse differentiation to find \(d(\arctan x)/dx\).

Find \(d(\arccos x)/dx\).

For \(f(x)=x^3\), find \((f^{-1})'(8)\).

For \(f(x)=2x+5\), find \((f^{-1})'(x)\).

For \(f(x)=x^3+x\), find \((f^{-1})'(2)\).

Find \(d(\arcsin(2x))/dx\) and state the input restriction.

Find \(d(\arctan(x^2))/dx\).

For \(f(x)=e^x\), use the inverse derivative formula to find \(d(\ln x)/dx\).

For \(f(x)=x^5+2x\), find \((f^{-1})'(3)\).

Find \(d(\arccos(1-2x))/dx\).

For \(f(x)=x+\sin x\), find \((f^{-1})'(0)\).

Show that if \(f'(a)=0\), the inverse derivative formula cannot be used at \(x=f(a)\).

For \(f(x)=x^3\), explain why \((f^{-1})'(0)\) is not finite.

Find \(d(\ln(\arctan x))/dx\), stating a domain condition.

A student says \(d(\arcsin x)/dx=1/\cos x\). Explain the error.

For \(f(x)=x^3+x\), prove \((f^{-1})'(x)>0\) for every real \(x\).

Derive the inverse derivative formula from \(f(y)=x\).