We are given the function $f(x) = x^4 - 2x^3 + 7x - \pi$, and we need to compute several derivatives of $f(x)$.

1. First derivative $f'(x)$

To find the first derivative, we apply the power rule to each term:

$f(x) = x^4 - 2x^3 + 7x - \pi$

• Derivative of $x^4$ is $4x^3$,
• Derivative of $-2x^3$ is $-6x^2$,
• Derivative of $7x$ is $7$,
• Derivative of $-\pi$ (a constant) is $0$.

Thus, the first derivative is:

$f'(x) = 4x^3 - 6x^2 + 7$

2. Second derivative $f''(x)$

Next, we take the derivative of $f'(x) = 4x^3 - 6x^2 + 7$:

• Derivative of $4x^3$ is $12x^2$,
• Derivative of $-6x^2$ is $-12x$,
• Derivative of $7$ is $0$.

Thus, the second derivative is:

$f''(x) = 12x^2 - 12x$

3. Third derivative $f'''(x)$

Now, we take the derivative of $f''(x) = 12x^2 - 12x$:

• Derivative of $12x^2$ is $24x$,
• Derivative of $-12x$ is $-12$.

Thus, the third derivative is:

$f'''(x) = 24x - 12$

4. Fourth derivative $f''''(x)$

Next, we differentiate $f'''(x) = 24x - 12$:

• Derivative of $24x$ is $24$,
• Derivative of $-12$ is $0$.

Thus, the fourth derivative is:

$f''''(x) = 24$

5. $f^{(50)}(x)$

Since the fourth derivative $f''''(x) = 24$ is a constant, all higher derivatives will be $0$. Thus, the 50th derivative is:

$f^{(50)}(x) = 0$

6. $f^{(5000)}(x)$

Similarly, since the fourth derivative is constant, the 5000th derivative will also be:

$f^{(5000)}(x) = 0$

Final Results:

1. $f'(x) = 4x^3 - 6x^2 + 7$
2. $f''(x) = 12x^2 - 12x$
3. $f'''(x) = 24x - 12$
4. $f''''(x) = 24$
5. $f^{(50)}(x) = 0$
6. $f^{(5000)}(x) = 0$

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Would you like any further clarifications on the derivatives or any specific details about the steps? Here are some related questions:

1. What is the general form of the derivative of a polynomial function?
2. How do constant terms affect derivatives?
3. What happens to the derivatives of higher-order terms (like $x^n$) as the order increases?
4. How does the power rule apply to negative exponents or fractional powers?
5. What happens when the degree of a polynomial exceeds the order of the derivative?

Tip: For any polynomial function, after taking the derivative multiple times, if the degree of the polynomial is reached, the subsequent derivatives will always be zero.