Derivative of e to the x squared

To understand the derivative of e to the x squared, we express it as d/dx (e^(x²)) or (e^(x²))'.

The derivative is 2x * e^(x²), indicating that the function is differentiable for all real numbers.

The key concepts include:

Exponential Function: e^(x²) is an exponential function of x².

Chain Rule: A differentiation technique necessary because the exponent is a function of x.

Exponential Growth: The function exhibits exponential growth as x increases.

The derivative of e^(x²) is expressed as d/dx (e^(x²)) or (e^(x²))'.

The formula used to differentiate e^(x²) is: d/dx (e^(x²)) = 2x * e^(x²)

The formula is valid for all real numbers x.

We can derive the derivative of e^(x²) using different methods, such as:

Using the Chain Rule

Using the First Principle We will demonstrate how the differentiation of e^(x²) results in 2x * e^(x²) using these methods:

Using the Chain Rule

To differentiate e^(x²) using the chain rule, we consider the exponent as a function of x. Let u = x², then f(u) = e^u and u' = 2x.

Therefore, by the chain rule: d/dx (e^(x²)) = d/du (e^u) * u' = e^(x²) * 2x = 2x * e^(x²)

By First Principle We can also prove the derivative using the First Principle, which expresses the derivative as the limit of the difference quotient.

Consider f(x) = e^(x²). Its derivative is: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h

Substitute f(x) = e^(x²), so f(x + h) = e^((x + h)²): f'(x) = limₕ→₀ [e^{((x + h)²}) - e^(x²)] / h

To solve this, we expand (x + h)² and apply the limit.

Ultimately, this results in the derivative 2x * e^(x²), aligning with the chain rule proof.

Higher-order derivatives are obtained by differentiating a function multiple times.

These derivatives provide insights into the behavior of the function, similar to how speed and acceleration describe a car's motion.

For e^(x²), higher-order derivatives can be expressed as follows:

The first derivative is 2x * e^(x²), representing the rate of change of the function.

The second derivative is derived from the first derivative, denoted f′′(x).

The third derivative, f′′′(x), follows from the second derivative, and this pattern continues.

For the nth derivative of e^(x²), we use fⁿ(x) to represent the nth derivative, indicating the change in the rate of change.

When x = 0, the derivative is 0 because e^(0²) = 1 and 2x = 0. For large positive or negative x, the derivative grows rapidly due to the exponential function's nature.

• Derivative: The derivative of a function represents how that function changes in response to changes in its input.

• Exponential Function: A mathematical function characterized by a constant base raised to a variable exponent, such as e^(x²).

• Chain Rule: A fundamental rule in calculus used to differentiate composite functions.

• Product Rule: A rule used in calculus to find the derivative of the product of two functions.

• Quotient Rule: A rule in calculus for differentiating the quotient of two functions.