Assessment: Methods of Differentiation

What does lim(x→a) f(x) = L mean?

What is d/dx[x⁷]?

Differentiate f(x) = x² + 3x, then find f'(2).

Differentiate f(x) = (x+1)(x+4) using the Product Rule, then find f'(0).

Differentiate f(x) = (x+3)/x using the Quotient Rule, then find f'(1).

Differentiate f(x) = (x+5)² using the Chain Rule, then find f'(3).

A company's profit function is P(x), for x units produced. What does the derivative P'(x) represent?

An engineer needs the slope of a curved cam profile given by an equation relating x and y that cannot easily be solved for y. Which technique should they use?

Why does the order of subtraction in the Quotient Rule's numerator (u'v − uv', not uv' − u'v) matter?

For the curve x² − xy + y² = 3, find dy/dx at the point (1, −1).