Derivative & Integral

Differentiate and integrate polynomials term by term, with exact rational coefficients and the power rule shown for every term.

Computed

The result and the term-by-term working are shown below.

Polynomial f(x)

degree 3

f(x) = x^3 + 2x + 3

Coefficients (ascending powers — leftmost is the constant term):

x^0

x^1

x^2

x^3

Each box is the coefficient of x^i. Integers, decimals or fractions.

d/dx

= 3x^2 + 2

1
Differentiate the polynomial
Apply the power rule d/dx [a·x^n] = n·a·x^(n-1) to x^3 + 2x + 3.
$\frac{d}{dx}\left[x^{3} + 2x + 3\right]$
2
Derivative of the constant term
d/dx [3] = 0 (constant drops out)
$\frac{d}{dx}\left[3\right] = 0$
3
Apply the power rule to the degree-1 term
$\frac{d}{dx}\left[2x\right] = 1 \cdot 2 = 2$
- d/dx [2·x] = 1·2·x^0 = 2
4
Apply the power rule to the degree-3 term
$\frac{d}{dx}\left[x^{3}\right] = 3 \cdot x^{2} = 3x^{2}$
- d/dx [1·x^3] = 3·1·x^2 = 3·x^2
5
Result
d/dx [x^3 + 2x + 3] = 3x^2 + 2
$f'(x) = 3x^{2} + 2$