Question

Find the third derivatives of
y
=
3
x
5
-
2
x
4
+
x
2
-
1

Answer 1

To find the third derivative of the function

\[
y = 3x^5 - 2x^4 + x^2 - 1,
\]

we will perform the differentiation step by step.

1. **First Derivative**:

\[
y' = \frac{dy}{dx} = \frac{d}{dx}(3x^5) - \frac{d}{dx}(2x^4) + \frac{d}{dx}(x^2) - \frac{d}{dx}(1)
\]
\[
y' = 15x^4 - 8x^3 + 2x.
\]

2. **Second Derivative**:

\[
y'' = \frac{d}{dx}(y') = \frac{d}{dx}(15x^4) - \frac{d}{dx}(8x^3) + \frac{d}{dx}(2x)
\]
\[
y'' = 60x^3 - 24x^2 + 2.
\]

3. **Third Derivative**:

\[
y''' = \frac{d}{dx}(y'') = \frac{d}{dx}(60x^3) - \frac{d}{dx}(24x^2) + \frac{d}{dx}(2)
\]
\[
y''' = 180x^2 - 48x.
\]

Thus, the third derivative of the given function is

\[
y''' = 180x^2 - 48x.
\]