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Differentiation rules

Differentials

The differentiation of a function of x, f(x), results in a new function describing how the tangent of the original curve varies with x. The symbol of this new function is f'(x) [f prime of x]

A common example is the differentiation of a curve describing the displacement of a body results in a curve describing its velocity:

$$\f(t) = at^2 + bt + c$$

where t is time, and a, b, c are constants

The differential of this quadratic can be determined from the general power rule:

If $$\f(x) = ax^n$$ then $$\f'(x) = an.x^(n-1)$$

so, $$\f'(t) = 2at + b$$

A further differentiation results in the acceleration of the body:

$$\f"(t) = 2a$$

Rules and examples of derivatives

Function $f(x)$ Derivative $f'(x)$ Rule name
$$mx + c$$ $$m$$ Constant multiple
$$x^n$$ $$nx^(n-1)$$ Power rule
$$u(x) + v(x)$$ $$u'(x) + v'(x)$$ Sum rule
$$u(x).v(x)$$ $$u'(x)v(x) + u(x)v'(x)$$ Product rule
$$ {u(x)}/{v(x)}$$ $$ {u'(x)v(x) - u(x)v'(x)}/{[v(x)]^2}$$ Quotient rule
$$y = f(u)$$ and $$u = u(x)$$ $${dy}/{dx} = {dy}/{du}⋅{du}/{dx}$$ Chain rule
$$e^{f(x)}$$ $$e^{f(x)} ⋅ f'(x)$$
$$ln(x)$$ $${1}/{x}$$
$$ln(f(x))$$ $${f'(x)} / {f(x)}$$
$$[f(x)]^n$$ $$n[f(x)]^{n-1} ⋅ f'(x)$$

Fundamental Theorem of Calculus

For f(x) to be continuous at x = c:

  • f must be defined at c: i.e. c must be an element of the domain of f
  • there must be an existing limit of f
  • the limit of f at c must be equal to the value of the function at c

Equations of tangents and normals

If $m$ is the gradient at point ($x_1, y_1$) on a curve, the equation of the tangent at that point is: $(y - y_1) = m(x - x_1)$

The normal to a point on a curve can be found from the fact that the normal to a line has a gradient which is $-1/m$, where $m$ is the gradient of the line.

Higher Order Derivatives

The first derivative of $f(x)$ is denoted as $f'(x)$ or ${dy}/{dx}$, and represents the slope (or gradient of the tangent to the curve of $f(x)$. e.g. If $s$ is the function of displacement, $s'(t) = {ds}/{dt} = v(t)$, the velocity of the particle as a function of time.

The second derivative of $f(x)$ is denoted as $f″(x)$ or ${d^2y}/{dx^2}$, and represents the slope (or gradient of the tangent to the curve of $f'(x)$. e.g. $s″(t) = {d^2s}/{dt^2} = {dv}/{dt} = a(t)$, the acceleration of the particle.

The third derivative of $f(x)$ is denoted as $f‴(x)$ or ${d^3y}/{dx^3}$, and represents the slope (or gradient of the tangent to the curve of $f″(x)$. e.g. $s‴(t) = {d^3s}/{dt^3} = {d^2v}/{dt^2} = {da}/{dt}$, the jerk, or rate of change of acceleration, of the particle.

Graphical interpretation of higher order derivatives

When the first derivative equals zero, the graph of $f(x)$ is parallel at that point to the $x$-axis, and this point is called a stationary point. This could be a local maximum (concave up) or minimum (concave down), or it could be an inflection point, and the graph continues up or down as before.

To determine whether the first derivative zero is a local maximum or minimum, the second derivative is taken to discover whether there is a change in sign (+ or -) of the first derivative from before the zero to after. If the sign does not change, then the zero was an inflection point.

If $f″(x) > 0$, $f(x)$ has a minimum at that point (i.e. the slope is changing from negative to positive). If $f″(x) < 0$, $f(x)$ has a maximum at that point (i.e. the slope is changing from positive to negative).


Associated Mathematicians:

  • Isaac Newton
  • Gottfried Leibnitz
  • Jakob Bernoulli
  • Johann Bernoulli (I)

Content © Andrew Bone. All rights reserved. Created : December 18, 2013 Last updated :December 13, 2015

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