If `l_(1)=(d)/(dx)(e^(sinx))`
`l_(2)lim_(hto0) (e^(sin(x+h))-e^(sinx))/(h)`
`l_(3)=inte^(sinx)cosxdx`
then which one of the following is correct?

Options

A. `l_(1)nel_(2)`
B. `(d)/(dx)(l_(3))=l_(2)`
C. `intl_(3)dx=l_(2)`
D. `l_(2)=l_(3)`

Explanation

To solve the problem, we need to evaluate the three expressions \( L_1 \), \( L_2 \), and \( L_3 \) and determine their relationships.

### Step 1: Calculate \( L_1 \)

Given:
\[ L_1 = \frac{d}{dx}(e^{\sin x}) \]

Using the chain rule:
\[ L_1 = e^{\sin x} \cdot \frac{d}{dx}(\sin x) = e^{\sin x} \cdot \cos x \]

### Step 2: Calculate \( L_2 \)

Given:
\[ L_2 = \lim_{h \to 0} \frac{e^{\sin(x+h)} - e^{\sin x}}{h} \]

This expression represents the definition of the derivative of \( e^{\sin x} \) at the point \( x \):
\[ L_2 = \frac{d}{dx}(e^{\sin x}) = e^{\sin x} \cdot \cos x \]

### Step 3: Calculate \( L_3 \)

Given:
\[ L_3 = \int e^{\sin x} \cos x \, dx \]

To solve this integral, we can use substitution. Let:
\[ t = \sin x \implies dt = \cos x \, dx \]

Then, the integral becomes:
\[ L_3 = \int e^t \, dt = e^t + C = e^{\sin x} + C \]

### Step 4: Differentiate \( L_3 \)

Now, we differentiate \( L_3 \):
\[ \frac{d}{dx}(L_3) = \frac{d}{dx}(e^{\sin x} + C) = e^{\sin x} \cdot \cos x \]

### Conclusion

Now we have:
- \( L_1 = e^{\sin x} \cdot \cos x \)
- \( L_2 = e^{\sin x} \cdot \cos x \)
- \( L_3 = e^{\sin x} + C \)

From the calculations:
- \( L_1 = L_2 \)
- \( \frac{d}{dx}(L_3) = L_1 = L_2 \)

Thus, the correct relationship is:
\[ \frac{d}{dx}(L_3) = L_2 \]

### Final Answer

The correct option is that \( L_1 = L_2 \) and \( \frac{d}{dx}(L_3) = L_2 \).

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