Find f^'(x) if f(x) is the given expression. (a) e^e (b) x^5 (c) x^v^5 (d) (√(5))

Find f^'(x) if f(x) is the given expression. (a) e^e ...

Find $f^{\prime}(x)$ if $f(x)$ is the given expression.
$\begin{array}{lll}\text { (a) } e^{e} & \text { (b) } x^{5} & \text { (c) } x^{v^{5}}\end{array}$
(d) $(\sqrt{5})^{x}$
(e) $x^{\left(x^{2}\right)}$

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is used when the variable appears both in the base and the exponent or in complicated products and quotients. By taking the natural logarithm of both sides, the differentiation process is simplified through properties of logarithms, facilitating the derivation of functions that would otherwise be difficult to differentiate directly.

Chain Rule

The chain rule is a fundamental tool for differentiating composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function, allowing for the systematic breakdown of complex derivatives.

Power Rule

The power rule is an essential differentiation technique for functions of the form x^n, where n is a constant. The derivative is calculated as n times x raised to the (n-1) power, making it straightforward to apply to monomials.

Exponential Function Differentiation with Constant Base

This concept deals with functions where the base of the exponential function is a constant and the exponent is the variable. The derivative of a function a^x is a^x multiplied by the natural logarithm of a, reflecting how the exponential growth rate is influenced by the base.

Differentiation of Constant Functions

When a function is constant, its output does not change with respect to any input variable. This means that its derivative is zero, regardless of how the constant is expressed, since there is no rate of change.