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Derivatives of Polynomials
Prerequisites:
The Power Rule
Linearity of the Derivative
Taking the derivative of polynomials is just applying linearity and the power rule. An example
d
d
x
(
2
x
3
+
3
x
2
+
x
+
5
)
=
2
d
(
x
3
)
d
x
+
3
d
(
x
2
)
d
x
+
d
(
x
)
d
x
+
d
(
5
)
d
x
=
2
(
3
x
2
)
+
3
(
2
x
)
+
1
=
6
x
2
+
6
x
+
1
\begin{aligned} &\frac{d}{dx}\left(2x^3 + 3x^2 + x + 5\right) \\ &= 2\frac{d(x^3)}{dx} + 3\frac{d(x^2)}{dx} + \frac{d(x)}{dx} + \frac{d(5)}{dx} \\ &= 2(3x^2) + 3(2x) + 1 \\ &= 6x^2 + 6x + 1 \end{aligned}
d
x
d
(
2
x
3
+
3
x
2
+
x
+
5
)
=
2
d
x
d
(
x
3
)
+
3
d
x
d
(
x
2
)
+
d
x
d
(
x
)
+
d
x
d
(
5
)
=
2
(
3
x
2
)
+
3
(
2
x
)
+
1
=
6
x
2
+
6
x
+
1