You can differentiate an expression, or find a partial derivative. For example, you can differentiate with respect to
x the following expression:
(6 + 3*x + 2*y + x*y) and obtain the result
3 + y. Differentiating the same expression with respect to
y gives the result
2 + x.
You can also find derivatives of higher order. For example, the instruction
diff(a*x^2 + b*x^3, x, 2) finds the second derivative with respect to
x of the expression
a*x^2 + b*x^3, giving the result
2*a + 6*b*x.
Algebra UltimaCalc can integrate many expressions, and tell you how it found the result. For example, the multiple-instruction line:
integrate(x*sin(x), x); how() finds the integral of
x*sin(x) to be
sin(x) - cos(x)*x. The instruction
how() shows the explanation:
"Integrate by parts using the equation:
"Integral( sin(x) * x ) = -cos(x) * x + Integral( cos(x) )
" cos(x) found in integral table"
It is easy to generate Taylor series. The following instruction finds the Taylor series for the expression
tan(x)^(1/2) in the vicinity of
x = pi/4:
taylor(tan(x)^(1/2), x, pi/4, h, 5). This finds terms up to the 5th order. The result is:
1 + h + 1/2*h^2 + 5/6*h^3 + 17/24*h^4 + 121/120*h^5 where
h = x - pi/4.