Mathematical Derivation of the Third Derivative of an Exponential Function

Fundaemental Principles of Exponential Differentiation

• The task involves calculating the third derivative of the exponential function $y = e^{5x}$. This process falls under the branch of calculus known as differential calculus, specifically focusing on higher-order derivatives.
• Natural Exponential Rule: The derivative of the natural exponential function $e^x$ with respect to $x$ is the function itself. Formally, this is represented as $\frac{d}{dx}(e^x) = e^x$.
• Chain Rule Integration: When the exponent is a function other than a single variable $x$, the Chain Rule must be applied. For a function $y = e^{u(x)}$, the derivative is $\frac{dy}{dx} = e^{u(x)} \times \frac{du}{dx}$.
• Higher-Order Derivatives: These represent the derivative of a derivative. The first derivative is denoted as $y'$ or $\frac{dy}{dx}$, the second derivative as $y''$ or $\frac{d^2y}{dx^2}$, and the third derivative as $y'''$ or $\frac{d^3y}{dx^3}$.

Problem Statement and Initial Variables

• Given Function: $y = e^{5x}$
• Target Objective: Find the third-order derivative, denoted as $y'''$.
• Inner Function ($u$): In this context, $u = 5x$.
• Derivative of Inner Function ($u'$): $\frac{d}{dx}(5x) = 5$.

Step-by-Step Derivation of the First Derivative ($y'$)

• To find the first derivative ($y'$), we apply the Chain Rule to the given function $y = e^{5x}$.
• Differentiate the outer function ($e^u$) while keeping the inner function ($5x$) unchanged, then multiply by the derivative of the inner function ($5$).
• Calculation:
• $y' = e^{5x} \times \frac{d}{dx}(5x)$
• $y' = e^{5x} \times 5$
• Resulting First Derivative: $y' = 5 \times e^{5x}$

Step-by-Step Derivation of the Second Derivative ($y''$)

• To find the second derivative ($y''$), we differentiate the first derivative $y' = 5 \times e^{5x}$ with respect to $x$.
• Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function. Formally, $\frac{d}{dx}(c \times f(x)) = c \times f'(x)$.
• Pull out the constant $5$ and differentiate $e^{5x}$.
• Calculation:
• $y'' = \frac{d}{dx}(5 \times e^{5x})$
• $y'' = 5 \times \frac{d}{dx}(e^{5x})$
• Applying the same exponential derivative rule used in Step 1:
• $y'' = 5 \times (5 \times e^{5x})$
• Resulting Second Derivative: $y'' = 25 \times e^{5x}$
• Alternative Notation: $y'' = 5^2 \times e^{5x}$

Step-by-Step Derivation of the Third Derivative ($y'''$)

• To find the third derivative ($y'''$), we differentiate the second derivative $y'' = 25 \times e^{5x}$ with respect to $x$.
• Pull out the constant factor of $25$ and differentiate the exponential term $e^{5x}$.
• Calculation:
• $y''' = \frac{d}{dx}(25 \times e^{5x})$
• $y''' = 25 \times \frac{d}{dx}(e^{5x})$
• Again, applying the derivative rule for $e^{5x}$ results in another factor of $5$.
• $y''' = 25 \times (5 \times e^{5x})$
• Resulting Third Derivative: $y''' = 125 \times e^{5x}$
• Alternative Notation: $y''' = 5^3 \times e^{5x}$

Generalization for the $n$-th Derivative

• Observation of the Pattern: Each subsequent differentiation task results in multiplying the current expression by the coefficient of the exponent ($5$).
• Pattern sequence:
• $n = 1 \rightarrow 5^1 \times e^{5x}$
• $n = 2 \rightarrow 5^2 \times e^{5x}$
• $n = 3 \rightarrow 5^3 \times e^{5x}$
• General Formula: For a function of the form $y = e^{ax}$, the $n$-th derivative is given by the formula:
• $y^{(n)} = a^n \times e^{ax}$
• Application to This Problem: Substituting $a = 5$ and $n = 3$ into the generalized formula:
• $y^{(3)} = 5^3 \times e^{5x}$
• $y^{(3)} = 125 \times e^{5x}$

Final Summary of Results

• The function provided is $y = e^{5x}$.
• The first derivative is $y' = 5 \times e^{5x}$.
• The second derivative is $y'' = 25 \times e^{5x}$.
• The third derivative, the solution requested, is $y''' = 125 \times e^{5x}$.
• This function is infinitely differentiable, and the magnitude of the coefficient will continue to grow by powers of $5$ with each additional derivative.