Differentiation: Definition and Fundamental Properties
Core ideas
The derivative is the limit of a difference quotient and represents instantaneous rate of change and slope of the tangent line. You should be able to compute a derivative from the definition when asked, and to recognize differentiability implies continuity but not conversely (corners, cusps, vertical tangents, and jumps break differentiability).
Master the power, constant, sum, product, and quotient rules, plus the derivatives of the standard functions. Notation fluency matters: $f'(x)$, $\frac{dy}{dx}$, and $y'$ all describe the same object. The tangent-line approximation is $y=f(a)+f'(a)(x-a)$, and linearization is the same idea framed as a local estimate.
Traps
- The product rule is not the product of derivatives; the quotient rule sign order is frequently reversed.
- Differentiability at a point requires the one-sided difference quotients to match; a continuous function can fail to be differentiable.
- When a question asks for the derivative from the limit definition, algebraic simplification before taking the limit is required.
- Higher derivatives: $f''$ describes concavity, not slope.
Worked example
Differentiate $f(x)=x^2\sin x$. By the product rule, $f'(x)=2x\sin x+x^2\cos x$. At $x=0$, $f'(0)=0$. The tangent line there is $y=0$, since $f(0)=0$. Notice the derivative exists everywhere because both factors are smooth.