Derivatives

Rates of change, the differentiation rules, and what they mean geometrically.

The derivative f′(x)f'(x) is the instantaneous rate of change of ff at xx — equivalently, the slope of the tangent line to the graph of ff at the point (x,f(x))(x, f(x)).
f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Rules you'll use every day

  • <strong>Power rule:</strong> ddxxn=nxn−1\frac{d}{dx} xn = n x{n-1}
  • <strong>Sum/difference:</strong> (f±g)′=f′±g′(f \pm g)' = f' \pm g'
  • <strong>Product:</strong> (fg)′=f′g+fg′(fg)' = f'g + fg'
  • <strong>Quotient:</strong> (fg)′=f′g−fg′g2\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}
  • <strong>Chain:</strong> ddxf(g(x))=f′(g(x))⋅g′(x)\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)

What a derivative tells you about a graph

  • f′(x)>0f'(x) > 0 on an interval ⇒\Rightarrow ff is increasing there.
  • f′(x)<0f'(x) < 0 on an interval ⇒\Rightarrow ff is decreasing there.
  • f′(c)=0f'(c) = 0 or undefined ⇒\Rightarrow cc is a <strong>critical point</strong> (candidate for a local max/min).
  • f′′(x)>0f''(x) > 0 ⇒\Rightarrow ff is concave up; f′′(x)<0f''(x) < 0 ⇒\Rightarrow concave down.