Differentiation and its applications

What you need to know

• The derivative dy/dx measures the gradient of the curve at a point, which is the rate at which y changes as x changes. A positive derivative means the curve is rising, a negative one means it is falling, and zero means it is momentarily flat.
• The basic rule is that if y = ax^n then dy/dx = anx^(n-1). Multiply by the power, then reduce the power by one. The derivative of a constant is zero, because a horizontal line has no gradient.
• The rule still works for negative and fractional powers, so rewrite roots and reciprocals as powers before differentiating. For y = 1/x^2 write y = x^(-2) and get dy/dx = -2x^(-3) = -2/x^3.
• The product rule for y = uv is dy/dx = u(dv/dx) + v(du/dx). Write down u, v, du/dx and dv/dx on separate lines first; candidates who try to do it in their heads lose a term.
• The quotient rule for y = u/v is dy/dx = [v(du/dx) - u(dv/dx)]/v^2. The order on the top matters because of the minus sign, so remember that the denominator term comes first.
• The chain rule for y = [f(x)]^n is dy/dx = n[f(x)]^(n-1) times f'(x): differentiate the outside, keep the inside unchanged, then multiply by the derivative of the inside. For y = (3x^2 - 1)^5, dy/dx = 5(3x^2 - 1)^4 times 6x = 30x(3x^2 - 1)^4.
• The standard derivatives you must know are: d/dx(sin x) = cos x, d/dx(cos x) = -sin x, d/dx(tan x) = sec^2 x, d/dx(e^x) = e^x and d/dx(ln x) = 1/x. These apply only when x is measured in radians.
• Stationary points occur where dy/dx = 0. Solve that equation for x, then substitute each x back into the original equation, not into the derivative, to find the corresponding y values.
• Classify a stationary point with the second derivative: if d2y/dx2 is positive the point is a minimum, if negative it is a maximum. The curve holds water at a minimum and spills it at a maximum, which is a reliable way to remember the signs.
• If the second derivative is zero at a stationary point, the test fails and you must examine the sign of dy/dx just before and just after the point. A change from negative to positive is a minimum, positive to negative a maximum, and no change is a point of inflexion.
• The gradient of the tangent at a point is the value of dy/dx there, and the gradient of the normal is the negative reciprocal of that value. Both lines pass through the same point, so use y - y1 = m(x - x1) with the appropriate m.
• Rates of change use the chain rule as dy/dt = (dy/dx)(dx/dt). A question about how fast a radius grows while the volume increases is solved by linking the two rates through the formula connecting the variables.
• For a small change, the approximation is small change in y is approximately (dy/dx) times small change in x. Percentage error questions use the same idea, with the percentage change in y being roughly (dy/dx)(change in x)/y times 100.
• In word problems on maximum area or minimum cost, form the expression to be optimised, use the given constraint to eliminate one variable, then differentiate with respect to the single remaining variable. Skipping the elimination step is why these questions go wrong.