Question

Solve . By chain rule to find derivatives of ex

Answer 1

$e^x$

Answer 2

To find the derivative of $e^x$ using the chain rule:

1. Identify the composite function:
   Let $y = e^x$.
   We can consider this as an outer function $f(u) = e^u$ and an inner function $u = g(x) = x$.

2. Differentiate the outer function with respect to $u$:
   $\frac{dy}{du} = \frac{d}{du}(e^u) = e^u$

3. Differentiate the inner function with respect to $x$:
   $\frac{du}{dx} = \frac{d}{dx}(x) = 1$

4. Apply the chain rule formula:
   The chain rule states that $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.
   Substitute the derivatives found in steps 2 and 3:
   $\frac{dy}{dx} = (e^u) \cdot (1)$

5. Substitute $u$ back with $x$:
   Since $u = x$, we replace $u$ with $x$:
   $\frac{dy}{dx} = e^x \cdot 1$
   $\frac{dy}{dx} = e^x$

Therefore, the derivative of $e^x$ is $e^x$.

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Explanation of the solution:
To find the derivative of $e^x$ using the chain rule, we consider $y = e^u$ where $u = x$. The derivative of the outer function $e^u$ with respect to $u$ is $e^u$. The derivative of the inner function $u=x$ with respect to $x$ is $1$. Applying the chain rule, $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = e^u \cdot 1$. Substituting $u=x$ back, we get $\frac{d}{dx}(e^x) = e^x$.