Question

For the equation $y = 3x^5 + 4x^4 - 3x^3 + x^2 - 2x + 1$, the value of $\frac{d^5y}{dx^5}$ is _ _ _. (answer in integer)

Hint:

For fifth derivative of a polynomial, only terms with power $5$ or more surviveHere, $3x^5$ gives $3 \times 5! = 360$

Answer 1

Correct Answer: 360

Step 1: Identify the given function.
\[ y = 3x^5 + 4x^4 - 3x^3 + x^2 - 2x + 1 \]

Step 2: Recall derivative rule.
For a polynomial term $ax^n$, the $n^{th}$ derivative becomes
\[ \frac{d^n}{dx^n}(ax^n)=a(n!) \]

Step 3: Identify the term contributing to fifth derivative.
Only the term $3x^5$ contributes to the fifth derivativeAll lower degree terms become zero after five differentiations

Step 4: Differentiate first time.
\[ \frac{dy}{dx}=15x^4+16x^3-9x^2+2x-2 \]

Step 5: Continue differentiation.
\[ \frac{d^2y}{dx^2}=60x^3+48x^2-18x+2 \]
\[ \frac{d^3y}{dx^3}=180x^2+96x-18 \]
\[ \frac{d^4y}{dx^4}=360x+96 \]
\[ \frac{d^5y}{dx^5}=360 \]

Step 6: Conclusion.
\[ \boxed{360} \]