Use the product rule: \((uv)' = u'v + uv'\). Let \(u = t^2\), \(v = e^t\).

["# Mastering the Product Rule in Differentiation: A Step-by-Step Guide with (u(t) = t^2) and (v(t) = e^t)", "When it comes to differentiating complex functions, few tools are as essential as the product rule. This fundamental theorem in calculus enables you to differentiate products of two differentiable functions seamlessly. In this article, we’ll explore how the product rule works, demonstrate its application using the functions (u(t) = t^2) and (v(t) = e^t), and explain why understanding this rule is crucial for advanced calculus and real-world problem solving.", "---", "## What Is the Product Rule?", "The product rule states that for two differentiable functions (u(t)) and (v(t)), the derivative of their product is given by:", "$$\n(uv)' = u'v + uv'\n$$", "In words: the derivative of (uv) is the derivative of (u) times (v), plus (u) times the derivative of (v). This formula saves time and reduces errors when tackling nonlinear functions.", "---", "## Step-by-Step Application with (u(t) = t^2) and (v(t) = e^t)", "Let’s apply the product rule by choosing:\n- (u(t) = t^2)\n- (v(t) = e^t)", "### Step 1: Differentiate (u(t)) and (v(t))", "First, compute the derivatives of (u) and (v):\n- (u'(t) = \frac{d}{dt}(t^2) = 2t)\n- (v'(t) = \frac{d}{dt}(e^t) = e^t)", "### Step 2: Apply the product rule formula", "Now plug the derivatives into the product rule formula:", "$$\n(uv)' = u'v + uv'\n$$", "Substitute the expressions:", "$$\n(t^2 \cdot e^t)' = (2t)(e^t) + (t^2)(e^t)\n$$", "### Step 3: Simplify the result", "Factor out (e^t) to make the expression cleaner:", "$$\n(t^2 e^t)' = e^t(2t + t^2) = e^t(t^2 + 2t)\n$$", "---", "## Final Answer", "$$\n\boxed{(t^2 e^t)' = e^t(t^2 + 2t)}\n$$", "---", "## Why Learning the Product Rule Matters", "Mastering the product rule is essential for advanced calculus, physics, engineering, and economics—any field where rates of change of composite functions are studied. It simplifies differentiation tasks involving error margins, optimization models, and dynamic systems.", "By practicing with clear examples like (u(t) = t^2) and (v(t) = e^t), you build fluency that translates directly to complex, real-world applications.", "---", "## More Practice Tips", "- Practice with other functions such as polynomials multiplied by exponentials or trigonometric functions.\n- Use online calculus tools or graphing software to visualize how the product rule impacts the curve’s slope.\n- Combine the product rule with other differentiation rules (like chain rule) for multi-layered functions.", "---", "## Conclusion", "The product rule—((uv)' = u'v + uv')—is more than a formula; it’s a reliable framework for tackling products of functions. With functions like (t^2) and (e^t), you gain a foundation to handle increasingly complicated derivatives. Keep practicing, stay curious, and watch your calculus skills grow stronger every day!", "---", "Keywords: product rule, calculus, derivatives, differentiation, ( (uv)' ), ( u = t^2 ), ( v = e^t ), math tutorial, calculus formula, product rule application, exponential function derivative"]