Mixed problems

Mixed problems test whether you choose the right method. Ask these questions in order:

• Can I simplify first? Inverse trig of a trig function, or a log of a product, often collapses into something simple.
• Is x in the power? Use logarithmic differentiation.
• Are x and y mixed? Use implicit differentiation.
• Is there a parameter? Divide the t-derivatives.

Simplify first example: tan⁻¹((1 – cos x)/sin x) for 0 < x < π. Using half-angle formulas, (1 – cos x)/sin x = 2sin²(x/2) / (2sin(x/2)cos(x/2)) = tan(x/2). So the expression is just x/2, and its derivative is ½.

Differentiating one function with respect to another: to find du/dv when both u and v depend on x, divide: du/dv = (du/dx) ÷ (dv/dx). Example: sin x with respect to cos x is cos x ÷ (–sin x) = –cot x.

Continuity versus differentiability with |...|: a sum of absolute values such as |x| + |x – 3| is continuous everywhere, but it has a sharp corner wherever one of the insides is zero, at x = 0 and at x = 3. It is not differentiable at those points.

Check the range of x given in the question: the simplification of an inverse trig expression often depends on it.