Derivative Product 29C3De

1. **State the problem:** Find the derivative of the function $$f(x) = e^x \cosh(x)$$. 2. **Recall the formula:** To differentiate a product of two functions, use the product rule: $$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$ 3. **Identify the functions:** Here, $$u(x) = e^x$$ and $$v(x) = \cosh(x)$$. 4. **Find the derivatives:** - $$u'(x) = \frac{d}{dx} e^x = e^x$$ - $$v'(x) = \frac{d}{dx} \cosh(x) = \sinh(x)$$ 5. **Apply the product rule:** $$f'(x) = u'(x)v(x) + u(x)v'(x) = e^x \cosh(x) + e^x \sinh(x)$$ 6. **Factor the expression:** $$f'(x) = e^x (\cosh(x) + \sinh(x))$$ 7. **Use the identity:** $$\cosh(x) + \sinh(x) = e^x$$ 8. **Final simplification:** $$f'(x) = e^x \cdot e^x = e^{2x}$$ **Answer:** $$\boxed{f'(x) = e^{2x}}$$