Write the first-principles definition of \(f'(x)\).

In \(\frac{f(x+h)-f(x)}{h}\), what does \(h\) represent?

For \(f(x)=x^2\), write \(f(x+h)\).

For \(f(x)=3x+1\), write the difference quotient \(\frac{f(x+h)-f(x)}{h}\).

Use first principles to differentiate \(f(x)=2x\).

Use first principles to differentiate \(f(x)=x+4\).

Use first principles to differentiate \(f(x)=x^2\).

Use first principles to differentiate \(f(x)=x^2+3\).

Use first principles to differentiate \(f(x)=x^3\).

Use first principles to find the derivative of \(f(x)=5-x^2\).

Use the alternative first-principles form to find \(f'(2)\) for \(f(x)=x^2+1\).

Explain why the factor \(h\) must be cancelled before setting \(h=0\) in a first-principles derivative.

Use first principles to differentiate \(f(x)=1/x\) for \(x\ne0\).

Use first principles to differentiate \(f(x)=\sqrt{x}\) for \(x>0\).

For \(f(x)=ax^2\), use first principles to find \(a\) if \(f'(3)=18\).

Use first principles to find \(f'(0)\) for \(f(x)=|x|\), or explain why it does not exist.

For \(f(x)=x^2+kx\), use first principles to find \(k\) if \(f'(1)=7\).

A student substitutes \(h=0\) into \(\frac{(x+h)^2-x^2}{h}\) and says the derivative is undefined. Diagnose the error.

Use first principles to show that the derivative of a constant function \(f(x)=c\) is zero.

Using first principles, explain why a corner in a graph can prevent differentiability.