The limit definition of the derivative states:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

For the given function:

$f(x) = (2x)^x$

we want to estimate $f'(3)$, so we compute:

$f'(3) \approx \frac{f(3+h) - f(3)}{h}$

Step 1: Compute $f(3)$

$f(3) = (2(3))^3 = 6^3 = 216$

Step 2: Compute $f(3+h)$ (for a small $h$)

$f(3+h) = (2(3+h))^{3+h}$

For estimation, let’s take $h = 0.001$:

$f(3+0.001) = (2(3.001))^{3.001}$

Now, we calculate:

$f'(3) \approx \frac{(2(3.001))^{3.001} - 216}{0.001}$

I will compute this numerically. Define function f(x) = (2x)^x def f(x): return (2 * x) ** x

Define values

x = 3 h = 0.001

Compute derivative approximation using limit definition

f_prime_approx = (f(x + h) - f(x)) / h f_prime_approxUsing the limit definition of the derivative with $h = 0.001$, we estimate:

$f'(3) \approx 603.90$

Would you like a more detailed breakdown of the calculations?

Related Questions:

1. How can we use logarithmic differentiation to find $f'(x)$ explicitly?
2. What happens to $f'(x)$ as $x$ increases?
3. Can we approximate $f'(3)$ more accurately using a smaller $h$?
4. How does the function $f(x) = (2x)^x$ behave for large $x$?
5. How can we use numerical differentiation techniques to estimate derivatives in general?

Tip:

For functions of the form $x^x$, taking the natural logarithm before differentiating simplifies the process!