Introduction to Derivatives

The derivative of a function represents its rate of change at a specific point.

\[f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]

Question 1

Find the derivative of $f(x) = x^2 + 3x + 5$.

To find the derivative, we use the power rule and the sum rule:

\[\begin{align*} f'(x) &= \frac{d}{dx}(x^2 + 3x + 5) \\ &= \frac{d}{dx}(x^2) + \frac{d}{dx}(3x) + \frac{d}{dx}(5) \\ &= 2x + 3 + 0 \\ &= 2x + 3 \end{align*}\]

Question 2

Find the derivative of $g(x) = e^x \sin(x)$ using the product rule.

Using the product rule: $(f \cdot g)’ = f’ \cdot g + f \cdot g’$

\[\begin{align*} \frac{d}{dx}[e^x \sin(x)] &= \frac{d}{dx}(e^x) \cdot \sin(x) + e^x \cdot \frac{d}{dx}(\sin(x)) \\ &= e^x \cdot \sin(x) + e^x \cdot \cos(x) \\ &= e^x[\sin(x) + \cos(x)] \end{align*}\]