What does the derivative \(f'(x)\) measure?

Write the limit definition of \(f'(x)\).

Differentiate \(f(x)=7\).

Differentiate \(f(x)=x\).

Differentiate \(f(x)=x^5\).

Differentiate \(y=3x^2\).

Differentiate \(f(x)=4x^3-2x+9\).

If \(s(t)=5t^2\) is position in metres, find the velocity \(v(t)\).

For \(f(x)=x^3-4x\), find \(f'(2)\).

Find the slope of the tangent to \(y=x^2+3x\) at \(x=-1\).

Find the equation of the tangent to \(y=x^2\) at \(x=3\).

If \(v(t)=12t-3t^2\), find the acceleration \(a(t)\).

Find where \(f(x)=x^3-3x\) has horizontal tangents.

For \(s(t)=t^3-6t^2+9t\), find when the velocity is zero.

Find \(k\) if \(f(x)=kx^2+3x\) has derivative \(11\) at \(x=2\).

For \(f(x)=ax^3+bx\), find conditions on \(a,b\) so \(f'(1)=5\) and \(f'(2)=14\).

Find \(c\) if the tangent to \(y=x^2+c\) at \(x=1\) passes through \((0,0)\).

A student says the derivative of \(x^4\) is \(x^3\). Identify the error and correct it.

Explain why differentiability at \(x=a\) implies continuity at \(x=a\).

A position graph has derivative \(0\) at one instant. Does that prove the object is at rest forever? Explain.