For a small input change delta x from x=a, use the tangent to estimate the output change: delta y (a)delta x. Thus f(a+delta x) (a)+f (a)delta x. The derivative is evaluated at the original value. Keep the sign of the change. This is an approximation, not an exact difference. Example: for y=sqrt(x), start at a=25. Then f (25)= 110. With delta x=0.2, sqrt(25.2) 5+ 0.210=5.02. For a percentage change, use 100 delta yy%. Calculus formulas must be recalled; check units for rates.

For a small input change \(\delta x\) from \(x=a\), use the tangent to estimate the output change: \(\delta y\approx\mathrm{f}^{\prime}(a)\delta x\). Thus \(\mathrm{f}(a+\delta x)\approx\mathrm{f}(a)+\mathrm{f}^{\prime}(a)\delta x\).

The derivative is evaluated at the original value. Keep the sign of the change. This is an approximation, not an exact difference.

Example: for \(y=\sqrt{x}\), start at \(a=25\). Then \(\mathrm{f}^{\prime}(25)=\dfrac1{10}\). With \(\delta x=0.2\), \(\sqrt{25.2}\approx5+\dfrac{0.2}{10}=5.02\).

For a percentage change, use \(100\dfrac{\delta y}{y}\%\). Calculus formulas must be recalled; check units for rates.

Small increments and approximations

Small increments and approximations