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Hint: Here in this question we will use the concept of composite function and differentiation to solve the question.

Complete step-by-step answer:

A composite function is a function that depends on another function. A composite function is created when a function is substituted in another function. In this question H(x) is a composite function because F(x) is being substituted in the function G(x).

Now, it is it given that F(x) = ${{\text{e}}^{\text{x}}}$, G(x) = ${{\text{e}}^{{\text{ - x}}}}$ and H(x) = G(F(x)). Now, finding H(x) by substituting F(x) in G(x). So,

H(x) = G (${{\text{e}}^{\text{x}}}$)

H(x) = ${{\text{e}}^{ - {{\text{e}}^x}}}$ …… (1)

Now we will differentiate equation (1) both sides with respect to x to find $\dfrac{{{\text{dH}}}}{{{\text{dx}}}}$. So,

$\dfrac{{{\text{dH}}}}{{{\text{dx}}}}{\text{ = }}\dfrac{{{\text{d(}}{{\text{e}}^{ - {{\text{e}}^{\text{x}}}}})}}{{{\text{dx}}}}$

As $\dfrac{{{\text{d(}}{{\text{e}}^{\text{x}}})}}{{{\text{dx}}}}{\text{ = }}{{\text{e}}^{\text{x}}}$. Applying chain rule in the above equation we get,

$\dfrac{{{\text{dH}}}}{{{\text{dx}}}}{\text{ = (}}{{\text{e}}^{ - {{\text{e}}^{\text{x}}}}})( - {{\text{e}}^{\text{x}}})(1)$

$ \Rightarrow $ $\dfrac{{{\text{dH}}}}{{{\text{dx}}}}{\text{ = - }}{{\text{e}}^{ - {{\text{e}}^{\text{x}}}}}{{\text{e}}^{\text{x}}}$ ……. (2)

Now we have to find the value of $\dfrac{{{\text{dH}}}}{{{\text{dx}}}}$ when x = 0. So, Putting x = 0 in the equation (2)

$ \Rightarrow $ $\dfrac{{{\text{dH}}}}{{{\text{dx}}}}{\text{ = - }}{{\text{e}}^{ - {{\text{e}}^0}}}{{\text{e}}^0}{\text{ = - }}{{\text{e}}^{ - 1}}{\text{ = - }}\dfrac{1}{{\text{e}}}{\text{ }}$

So, $\dfrac{{{\text{dH}}}}{{{\text{dx}}}}{\text{ = - }}\dfrac{1}{{\text{e}}}$ i.e. option (C) is the correct answer.

Note: While solving such problems, always substitute the correct function in another function. For example, if it is given F(G(x)) then substitute function G(x) in the function F(x) instead of doing the reverse which leads to incorrect answers. Also, perform differentiation properly to remove any error. Also, double check your answer to ensure that there is no mistake in the solution.

Complete step-by-step answer:

A composite function is a function that depends on another function. A composite function is created when a function is substituted in another function. In this question H(x) is a composite function because F(x) is being substituted in the function G(x).

Now, it is it given that F(x) = ${{\text{e}}^{\text{x}}}$, G(x) = ${{\text{e}}^{{\text{ - x}}}}$ and H(x) = G(F(x)). Now, finding H(x) by substituting F(x) in G(x). So,

H(x) = G (${{\text{e}}^{\text{x}}}$)

H(x) = ${{\text{e}}^{ - {{\text{e}}^x}}}$ …… (1)

Now we will differentiate equation (1) both sides with respect to x to find $\dfrac{{{\text{dH}}}}{{{\text{dx}}}}$. So,

$\dfrac{{{\text{dH}}}}{{{\text{dx}}}}{\text{ = }}\dfrac{{{\text{d(}}{{\text{e}}^{ - {{\text{e}}^{\text{x}}}}})}}{{{\text{dx}}}}$

As $\dfrac{{{\text{d(}}{{\text{e}}^{\text{x}}})}}{{{\text{dx}}}}{\text{ = }}{{\text{e}}^{\text{x}}}$. Applying chain rule in the above equation we get,

$\dfrac{{{\text{dH}}}}{{{\text{dx}}}}{\text{ = (}}{{\text{e}}^{ - {{\text{e}}^{\text{x}}}}})( - {{\text{e}}^{\text{x}}})(1)$

$ \Rightarrow $ $\dfrac{{{\text{dH}}}}{{{\text{dx}}}}{\text{ = - }}{{\text{e}}^{ - {{\text{e}}^{\text{x}}}}}{{\text{e}}^{\text{x}}}$ ……. (2)

Now we have to find the value of $\dfrac{{{\text{dH}}}}{{{\text{dx}}}}$ when x = 0. So, Putting x = 0 in the equation (2)

$ \Rightarrow $ $\dfrac{{{\text{dH}}}}{{{\text{dx}}}}{\text{ = - }}{{\text{e}}^{ - {{\text{e}}^0}}}{{\text{e}}^0}{\text{ = - }}{{\text{e}}^{ - 1}}{\text{ = - }}\dfrac{1}{{\text{e}}}{\text{ }}$

So, $\dfrac{{{\text{dH}}}}{{{\text{dx}}}}{\text{ = - }}\dfrac{1}{{\text{e}}}$ i.e. option (C) is the correct answer.

Note: While solving such problems, always substitute the correct function in another function. For example, if it is given F(G(x)) then substitute function G(x) in the function F(x) instead of doing the reverse which leads to incorrect answers. Also, perform differentiation properly to remove any error. Also, double check your answer to ensure that there is no mistake in the solution.

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