Numerical differentiation and numerical integration are fundamental techniques in computational mathematics used to approximate derivatives and integrals, respectively, of functions when analytical solutions are not feasible or practical to obtain.

Numerical Differentiation:

Numerical differentiation involves estimating the derivative of a function at a point using discrete data points. Several common methods are used for numerical differentiation:

1. Forward Difference Method: It approximates the derivative of a function

   $$

   f(x) at a point
   $$

   x using the formula:
   $$

   f′(x)≈hf(x+h)−f(x)​ where
   $\mathrm{<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">h\; </span></span></span></span>is\; a\; small\; increment.</span>}$

2. Backward Difference Method: Similar to the forward difference method, but it computes the derivative using values of the function at 
   $$x

   x−h:
   ${}^{}$

   f′(x)≈hf(x)−f(x−h)​

3. Central Difference Method: It provides a more accurate approximation by using values on both sides of the point

   $$

   x:
   ${}^{}$

   f′(x)≈2hf(x+h)−f(x−h)​

Numerical Integration:

Numerical integration, also known as quadrature, involves approximating the definite integral of a function over a given interval. Common methods for numerical integration include:

1. Trapezoidal Rule: This method approximates the area under a curve by dividing the interval into trapezoids. For a function

   $$

   f(x) over the interval
   $$

   [a,b], the trapezoidal rule is given by:
   ${}_{}^{}$

   ∫ab​f(x)dx≈2h​[f(a)+2∑i=1n−1​f(xi​)+f(b)] where
   $ℎ=\frac{}{}$

   h=nb−a​ and
   ${}_{}$

   xi​=a+ih.

2. Simpson’s Rule: This method uses quadratic interpolation to approximate the function between each pair of points and integrates the resulting parabolas. For a function

   $$

   f(x) over the interval
   $$

   [a,b], Simpson’s rule is given by:
   ${}_{}^{}$

   ∫ab​f(x)dx≈3h​[f(a)+4∑i=1n/2−1​f(x2i​)+2∑i=1n/2​f(x2i−1​)+f(b)] where
   $ℎ=\frac{}{}$

   h=nb−a​,
   $$

   n is even, and
   ${}_{}$

   xi​=a+ih.

These methods are invaluable for approximating derivatives and integrals when analytic solutions are not available or when dealing with complex functions. The choice of method depends on factors such as the nature of the function, the desired accuracy, and computational efficiency