The most important property of the natural exponential function $\ e^x$ is that its derivative is the function itself: $\frac{d}{dx} e^x = e^x$

• This means that the rate of change of $\ e^x$ with respect to $\ x$ is exactly $\ e^x .$
• This property is unique to the base $\ e \approx 2.718$, making it the natural choice in many mathematical and scientific contexts.

For a general exponential function with base $\ a$, where $\ a > 0$, the derivative is given by: $\frac{d}{dx} a^x = a^x \ln(a)$

• Here, $\ \ln(a)$ is the natural logarithm of $\ a$, which acts as a constant multiplier.
• This formula shows that while $\ a^x$ has a similar form to $\ e^x ,$ its rate of change is scaled by the factor $\ \ln(a) .$

$\frac{d}{dx} e^{u(x)} = e^{u(x)} \cdot u'(x)$

• First, differentiate the exponent $\ u(x)$ to find $\ u'(x)$.
• Then multiply by the original function $\ e^{u(x)} .$

$\frac{d}{dx} a^{u(x)} = a^{u(x)} \ln(a) \cdot u'(x)$

• Again, apply the chain rule by first differentiating $\ u(x)$ and then multiplying by $\ a^{u(x)} \ln(a) .$

Example 1: Derivative of $\ y = e^{3x}$

• Solution:
• Let $\ u(x) = 3x .$
• Differentiate$\ u(x)$ with respect to $\ x : \ u'(x) = 3 .$
• Apply the chain rule: $\frac{dy}{dx} = e^{3x} \cdot 3 = 3e^{3x}$

Example 2: Derivative of $\ y = 5^x$

• Solution:
• The base is $\ a = 5$, so: $\frac{dy}{dx} = 5^x \ln(5)$

Example 3: Derivative of $\ y = 2e^{-x^2}$

• Solution:
• Here, $\ u(x) = -x^2 .$
• Differentiate $\ u(x)$ with respect to $\ x \ : \ u'(x) = -2x$.
• Apply the chain rule: $\frac{dy}{dx} = 2e^{-x^2} \cdot (-2x) = -4x e^{-x^2}$

Example 4: Derivative of $\ y = 3x \cdot e^{2x}$

This function involves a product of two functions, so you'll need the product rule.

• Solution:
• Let $\ u(x) = 3x$ and $\ v(x) = e^{2x} .$
• The product rule is $\ \frac{d}{dx}[uv] = u'v + uv' .$
• Differentiate $\ u(x) = 3x$ to get $\ u'(x) = 3 .$
• Differentiate $\ v(x) = e^{2x} \ using\ the\ chain\ rule\ to\ get\ \ v'(x) = 2e^{2x} .$
• Apply the product rule: $\frac{dy}{dx} = 3 \cdot e^{2x} + 3x \cdot 2e^{2x} = 3e^{2x} + 6x e^{2x} = 3e^{2x}(1 + 2x)$

• The derivative of $\ e^x$ is unique because it is equal to the function itself.
• The derivative of $\ a^x$ includes a natural logarithm term, scaling the rate of change.
• When dealing with more complex exponents, the chain rule is essential.
• Mastering these derivatives is crucial for solving problems involving exponential growth, decay, and other natural phenomena.