Functions and Derivatives
Functions and derivatives
A function is a rule with a domain; before differentiating or solving, state the domain forced by denominators, even roots, and logarithms. Know composition, inverse functions (domain and range swap), symmetry, periodicity, and transformations of graphs. The derivative is the limit of a difference quotient; interpret f′(x₀) as the tangent slope and use it for linear approximation. Master power, product, quotient, exponential, logarithmic, and chain rules. For monotonicity, the sign of f′ decides intervals; local extrema require a sign change or a second-derivative test, and on a closed interval compare values at critical points and endpoints. Concavity and f″ help with inflection and inequalities. Differentiability implies continuity, but continuity does not imply differentiability, as |x| at 0 shows.
Traps include dropping the inner derivative in a chain rule, writing the derivative of aˣ as x a^{x−1}, assuming f′(x₀)=0 guarantees an extremum, and ignoring endpoint candidates. Also beware of cancelling factors that vanish, and of concluding f is increasing from f′(x)≥0 without checking where equality occurs.
Worked example: for f(x)=x ln x on x>0, f′(x)=ln x+1. Setting f′=0 gives x=e^{−1}. The sign chart is negative on (0,e^{−1}) and positive afterward, so the minimum value is f(e^{−1})=−1/e. Rule of thumb: domain first, derivative second, sign chart third, and always compare endpoints when the interval is closed.