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Chris Fox's Engineering Section

Calculus: Differentiation

Calculus, whose name derives from the Latin for "small pebble", is the branch of mathematics concerned with rates of variation. It consists of two parts: differentiation, which is analogous to finding the gradient of a line on a graph; and integration, analogous to finding the area under that line.

The gradient, or derivative, of a line, is denoted in the form dy/dx. This denotes the rate of change of y with respect to x, where y is a function of x. The function can be inserted in place of y.

One way to calculate the derivative is by First Principles Differentiation. The formula for this is dy/dx=limh → 0 {y(x+h)-y(x)}/h. The formulae for the values of y at points x+h and x are inserted and the equation simplified. This gives the change in y divided by the change in x for a finite distance. On a curved graph, however, the gradient is constantly changing, which means the gradient at a particular point applies only at that point. To make the formula work, therefore, h is then set to 0.

If the function to be differentiated is complex, the process can be simplified by using the Chain Rule, which states that (dy/dx)=(dy/du)(du/dx). Write the formula in the form y=f(u), where u=f(x), then differentiate with respect to u. Next, replace u with f(x), then multiply the result by du/dx.

The product of two functions of x can be differentiated using the Product Rule. If the functions are termed u and v, d(uv)/dx=u(dv/dx)+v(du/dx).

If the function to be differentiated consists of one function divided by another, the Quotient Rule can be used: d(u/v)/dx={v(du/dx)-u(dv/dx)}/v².

Where the dependent variable is a function of more than one other variable, e.g. z=f(x,y), it may be differentiated with respect to one variable, with the others treated as constants. This process is called partial differentiation, and the symbol ∂ is used in place of the d.

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