If `f(x) = int_(1//x^(2))^(2) cos sqrtt dt` then f'(1) is equal to

Options

A. cos 1
B. 2cos 1
C. 4 cos 1
D. None of these

Answer

To solve the problem, we need to find the derivative of the function defined by the integral:

\[ f(x) = \int_{\frac{1}{x^2}}^{2} \cos(\sqrt{t}) \, dt \]

and then evaluate \( f'(1) \).

### Step 1: Differentiate \( f(x) \)

Using the Fundamental Theorem of Calculus and the Leibniz rule for differentiation under the integral sign, we have:

\[ f'(x) = \frac{d}{dx} \left( \int_{\frac{1}{x^2}}^{2} \cos(\sqrt{t}) \, dt \right) \]

According to the Leibniz rule, we differentiate the integral with respect to \( x \):

\[ f'(x) = \cos(\sqrt{2}) \cdot 0 - \cos\left(\sqrt{\frac{1}{x^2}}\right) \cdot \frac{d}{dx}\left(\frac{1}{x^2}\right) \]

### Step 2: Calculate the derivative of the lower limit

The derivative of the lower limit \( \frac{1}{x^2} \) is:

\[ \frac{d}{dx}\left(\frac{1}{x^2}\right) = -\frac{2}{x^3} \]

### Step 3: Substitute back into \( f'(x) \)

Now substituting back, we have:

\[ f'(x) = -\cos\left(\frac{1}{x}\right) \cdot \left(-\frac{2}{x^3}\right) = \frac{2 \cos\left(\frac{1}{x}\right)}{x^3} \]

### Step 4: Evaluate \( f'(1) \)

Now we need to evaluate \( f'(1) \):

\[ f'(1) = \frac{2 \cos\left(\frac{1}{1}\right)}{1^3} = 2 \cos(1) \]

### Conclusion

Thus, the value of \( f'(1) \) is:

\[ f'(1) = 2 \cos(1) \]