Let's analyze each statement to determine its correctness.

Statement Analysis

1. Statement 1: $y = a^x \Rightarrow y^{(5)} = a^x \ln^5 a$

   • This is a fifth derivative problem. For $y = a^x$, the derivatives are $y' = a^x \ln a$, $y'' = a^x (\ln a)^2$, and so on. The fifth derivative would indeed be $y^{(5)} = a^x (\ln a)^5$, so this statement is correct.

2. Statement 2: $y = \sin x \Rightarrow y^{(7)} = \sin \left(x + \frac{7\pi}{2}\right)$

   • The derivatives of $\sin x$ cycle every four terms: $y' = \cos x$, $y'' = -\sin x$, $y^{(3)} = -\cos x$, and $y^{(4)} = \sin x$. The seventh derivative $y^{(7)}$ would correspond to $\cos x$ (since $7 \mod 4 = 3$), not a shifted sine function. This statement is incorrect.

3. Statement 3: $y = e^x \Rightarrow y^{(14)} = e^x$

   • For any power derivative of $e^x$, the result is still $e^x$. Thus, this statement is correct.

4. Statement 4: $y = 5x^4 + 3x^3 \Rightarrow d^4 y = 120x dx^4$

   • The fourth derivative of $5x^4$ is $120$, and for $3x^3$, the fourth derivative is zero. So $y^{(4)} = 120$, a constant, not $120x$. This statement is incorrect.

5. Statement 5: $y = \sin x \Rightarrow y^{(8)} = -\cos x$

   • Since derivatives of $\sin x$ cycle every four terms, the eighth derivative $y^{(8)}$ would return to $\sin x$, not $-\cos x$. This statement is incorrect.

6. Statement 6: $y = 17x^7 \Rightarrow y^{(9)} = 0$

   • The ninth derivative of a polynomial of degree 7 is indeed zero because we run out of terms after the seventh derivative. This statement is correct.

7. Statement 7: $y = 4x^5 \Rightarrow y^{(4)} = 480x$

   • Taking the fourth derivative of $4x^5$ yields $y^{(4)} = 480x$, which is correct. This statement is correct.

8. Statement 8: $y = 4x^7 \Rightarrow d^6 y = 0$

   • The sixth derivative of $4x^7$ is not zero; rather, it would yield a non-zero value (as $d^6 y$ still involves $x$ terms). This statement is incorrect.

9. Statement 9: $y = 5x^4 + 2x^2 \Rightarrow y^{(3)} = 120x$

   • For $y = 5x^4 + 2x^2$, the third derivative yields $y^{(3)} = 120x$. This statement is correct.

10. Statement 10: $y = \sin x \Rightarrow d^7 y = \sin x dx$

    • This statement is nonsensical as it misuses the notation. The seventh derivative of $\sin x$ would not yield $\sin x dx$. This statement is incorrect.

11. Statement 11: $y = 4x^7 \Rightarrow d^7 y = -\cos x dx^7$

    • This statement is also nonsensical and incorrectly mixes trigonometric functions and derivatives of polynomials. This statement is incorrect.

Correct Statements

The correct statements are:

1. Statement 1
2. Statement 3
3. Statement 6
4. Statement 7
5. Statement 9

Would you like a more detailed breakdown of each derivative step-by-step?

Related Questions

1. What is the general formula for the nth derivative of $a^x$?
2. Why do trigonometric functions cycle in their derivatives?
3. How does the degree of a polynomial affect the order of its zero derivatives?
4. What are common mistakes when differentiating exponential and trigonometric functions?
5. How does one interpret the notation $d^n y$ and $y^{(n)}$?

Tip

Always simplify derivative terms and carefully follow cyclic properties of trigonometric derivatives to avoid errors.