Techniques of differentiation

The derivative is the limit of a difference quotient. The chain rule tracks nested dependence: $d[f(g(x))]/dx=f\prime(g(x))g\prime(x)$. For implicit equations, y depends on x, so differentiating a y-term introduces $dy/dx$. Limits describe approaching a point even when the value there is separately defined.

Limits and continuity

If f(x) approaches L as x approaches a, write $\lim_{x\to a}f(x)=L$. For $f(x)=x^2-x+2$, the limit as x approaches 2 is 4.

A limit can exist where the function is undefined. In radians, $\sin x/x$ is undefined at zero but has limit 1 there; its graph has a missing point at $(0,1)$. Continuity at a requires a defined value and $\lim_{x\to a}f(x)=f(a)$. The notes’ classroom continuity assumption applies on the interval under consideration, not at excluded inputs.

Approach from both sides

For the source function $f(x)=x^2-x+2$, the two printed value lists approach $x=2$ from opposite sides:

x from the left	f(x)	x from the right	f(x)
1.0	2.000000	3.0	8.000000
1.5	2.750000	2.5	5.750000
1.8	3.440000	2.2	4.640000
1.9	3.710000	2.1	4.310000
1.95	3.852500	2.05	4.152500
1.99	3.970100	2.01	4.030100
1.995	3.985025	2.005	4.015025
1.999	3.997001	2.001	4.003001

Both lists approach 4. Limits underpin calculus, including tangent gradients. The derivative is itself a function: it maps an input x to the gradient of the original function’s tangent at that input, wherever that derivative exists.

A secant approaches a tangent

For $P=(x,f(x))$ and $Q=(x+h,f(x+h))$, the secant gradient is $[f(x+h)-f(x)]/h=\Delta y/h$, for $h\ne0$. If its limit exists as $h\to0$, that limit is the tangent gradient $f^{\prime}(x)=dy/dx$. The source uses $\delta x$ instead of h and calls the tangent AB. Continuity alone does not guarantee a derivative; the finite limit must exist.

The source’s page 5 QR code points to an exploration of the limiting process. This preserves the original resource pointer; the external activity is not claimed as reviewed. Appendix Examples A–C on PDF 17–20 are included below with their first-principles working and calculator-limit qualifications.