Differences between revisions 4 and 5

Differential Calculus

Besides the examples on this page, please see the discussion in ["BasicCalculus"].

Functions

Piecewise fcns, polynomials, exponential, logs, trig and hyperboic trig functions.

SAGE can compute \lim_{x\rightarrow 0}\frac{\sin(x)}{x}:

sage: limit(sin(x)/x,x=0)
1

SAGE can differentiate x^2\log(x+a) and \tan^{-1}(x)=\arctan(x):

sage: diff(x^2 * log(x+a), x)
2*x*log(x + a) + x^2/(x + a)
sage: derivative(atan(x), x)
1/(x^2 + 1)

SAGE can verify the product rule

sage: function('f, g')
(f, g)
sage: diff(f(t)*g(t),t)
f(t)*diff(g(t), t, 1) + g(t)*diff(f(t), t, 1)

the quotient rule

sage: diff(f(t)/g(t), t)
diff(f(t), t, 1)/g(t) - (f(t)*diff(g(t), t, 1)/g(t)^2)

and linearity:

sage: diff(f(t) + g(t), t)
diff(g(t), t, 1) + diff(f(t), t, 1)
sage: diff(c*f(t), t)
c*diff(f(t), t, 1)

...

(Some schools teach this topic as part of integral calculus.)

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+=== Laws ===

SAGE can verify the product rule

{{{
sage: function('f, g')
(f, g)
sage: diff(f(t)*g(t),t)
f(t)*diff(g(t), t, 1) + g(t)*diff(f(t), t, 1)
}}}
the quotient rule

{{{
sage: diff(f(t)/g(t), t)
diff(f(t), t, 1)/g(t) - (f(t)*diff(g(t), t, 1)/g(t)^2)
}}}

and linearity:

{{{
sage: diff(f(t) + g(t), t)
diff(g(t), t, 1) + diff(f(t), t, 1)
sage: diff(c*f(t), t)
c*diff(f(t), t, 1)
}}}