State the quotient rule for \(\frac{f(x)}{g(x)}\).

What condition must hold before differentiating \(f(x)/g(x)\) at a point?

Differentiate \(\frac{x}{\sin x}\).

Differentiate \(\frac{e^x}{x}\).

Differentiate \(\frac{x^2}{x+1}\).

Differentiate \(\frac{\sin x}{x}\).

Differentiate \(\frac{x^3+1}{x^2}\) using the quotient rule.

Differentiate \(\frac{\ln x}{x}\) for \(x>0\).

Find \(f'(1)\) for \(f(x)=\frac{x^2+1}{x+2}\).

Differentiate \(\frac{x^2-1}{x-1}\) for \(x\ne1\), and compare with simplifying first.

Differentiate \(\frac{x\cos x}{x+1}\).

Explain the sign order in the quotient rule using \((f/g)'\).

Differentiate \(\frac{x^2\sin x}{e^x}\).

Differentiate \(\frac{x^2+1}{\sin x}\).

Find \(a\) if \(f(x)=\frac{x+a}{x+1}\) has \(f'(0)=3\).

Find all \(x\ne0\) where \(f(x)=\frac{x^2+1}{x}\) has horizontal tangent.

Find \(k\) if \(f(x)=\frac{kx}{x+2}\) has \(f'(2)=1\).

A student differentiates \(\frac{x^2}{x+1}\) as \(\frac{2x}{1}\). Diagnose the error.

Show how the quotient rule follows from writing \(f/g=f\cdot g^{-1}\).

Explain why the derivative of \(\frac{x^2-1}{x-1}\) at \(x=1\) does not exist even though the simplified derivative is \(1\).