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
.