Find \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) where \(y(x)\) is defined by (a) \(3 x^{2}-\mathrm{e}^{2 x}\) (b) \(\sin 3 x+\cos x\) (c) \(\sqrt{x}\) (d) \(\mathrm{e}^{x}+\mathrm{e}^{-x}\) (e) \(1+x+x^{2}+\ln x\)

The Power Rule is a fundamental technique in calculus used to find the derivative of functions that are in the form of powers of the variable. It states that if you have a function of the form
\( f(x) = ax^n \),
then the derivative of that function with respect to
\( x \) is
\( f'(x) = nax^{n-1} \).
This rule simplifies the process of differentiation by directly providing the derivative without the need for complex calculations. For example, applying the Power Rule to \( x^2 \),
the derivative is
\( 2x \),
since you bring down the exponent as a coefficient and decrease the exponent by one.

In the exercise given, we see the Power Rule applied to terms like
\( 3x^2 \)
and \( x^2 \), leading to derivatives of
\( 6x \) and \( 2x \), respectively. The Power Rule is also adapted to tackle terms with fractions or negative exponents, such as
\( x^{1/2} \)
which becomes
\( \frac{1}{2}x^{-1/2} \) when differentiated. It’s straightforward, powerful, and is a foundational tool in the arsenal of calculus techniques.

The Chain Rule is another essential tool in calculus, particularly useful for differentiating composite functions. When you have a function within a function, or 'nested' functions,
\( (f \circ g)(x) = f(g(x)) \),
the Chain Rule helps you to unravel this nesting. The rule states that the derivative of the composite function \( f(g(x)) \) is
\( f'(g(x)) \cdot g'(x) \).
This means you first find the derivative of the outer function with respect to the inner function and then multiply it by the derivative of the inner function with respect to x.

In our exercises, the Chain Rule is utilized in terms like
\( \mathrm{e}^{2x} \)
and \( \sin(3x) \), where the exponent and the argument of the sine function are functions of
\( x \) themselves. Using the Chain Rule allows us to systematically find the derivatives of these more complex expressions by acknowledging their composite nature and differentiating in stages.

Differentiating trigonometric functions follows specific rules that correspond to each function. The derivatives of basic trigonometric functions are crucial to know for calculus students.
For example, the derivative of \( \sin(x) \) is
\( \cos(x) \),
and the derivative of \( \cos(x) \) is
\( -\sin(x) \).
These are straightforward when the functions are alone, but when combined with other functions or multiplied by constants, then applying the Chain Rule often becomes necessary.

In the exercises provided, we differentiate \( \sin(3x) + \cos(x) \). Here, derivates of \( \sin(3x) \) and \( \cos(x) \) are found using properties of trigonometric derivatives combined with the Chain Rule. Understanding these properties is essential for swiftly managing derivatives involving trigonometric functions, which appear frequently in various fields of science and engineering.