Solving differential equations



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Introduction to Algebraic manipulations

Messy limit calculations can be avoided, in certain cases, because of differentiation rules which allow one to find derivatives via algebraic manipulation; rather than by direct application of Newton's difference quotient. One should not infer that the definition of derivatives, in terms of limits, is unnecessary. Rather, that definition is the means of proving the following "powerful differentiation rules"; these rules are derived from the difference quotient.

Constant Rule: The derivative of any constant is zero.
- Constant Multiple Rule: If c is some real number; then, the derivative of $c f (x)$ equals c multiplied by the derivative of f(x) (a consequence of linearity below)
Linearity: (af + bg)' = af ' + bg' for all functions f and g and all real numbers a and b.
General Power Rule (Polynomial rule): If $f (x) = x r$ , for some real number r; $f'(x) = r x r - 1$ .
Product Rule: (fg)' = f 'g + fg' for all functions f and g.
Quotient Rule: (f/g)' = (f 'g - fg')/ $(g 2)$ unless g is zero.
Chain Rule: If f(x) = h(g(x)), then f '(x) = h'[g(x)] * g'(x).
Inverse functions and differentiation: If $y = f (x)$ , $x = f - 1 (y)$ , and f(x) and its inverse are differentiable, then for cases in which $\Delta x \ne 0$ when $\Delta y \ne 0$ , $d y / d x = 1 / (d x / d y)$
Derivative of one variable with respect to another when both are functions of a third variable: Let $x = f (t)$ and $y = g (t)$ . Now $Δ y / Δ x = (Δ y / Δ t) / (Δ x / Δ t).$
Implicit differentiation: If $f (x, y) = 0$ is an implicit function, we have: dy/dx = - (∂f / ∂x) / (∂f / ∂y).

In addition, the derivatives of some common functions are useful to know. See the table of derivatives.

As an example, the derivative of

$f(x) = 2x^4 + \sin (x^2) - \ln (x)\;e^x + 7$

$f'(x) = 8x^3 + 2x\cos (x^2) - (1/x)\;e^x - \ln (x)\;e^x.$

Solving linear equations