Given that p(x) = (5e^x+10x⁵+20x³+100x²+5x+20) ⋅ (10x⁵+40x³+20x²+4x+10), find p′(0) without computing p′(x).

Verified step by step guidance

1. Step 1: Recognize that p(x) is a product of two functions, say f(x) = 5e^x + 10x^5 + 20x^3 + 100x^2 + 5x + 20 and g(x) = 10x^5 + 40x^3 + 20x^2 + 4x + 10.
2. Step 2: Use the product rule for differentiation, which states that if p(x) = f(x)g(x), then p'(x) = f'(x)g(x) + f(x)g'(x).
3. Step 3: Evaluate f(0) and g(0) by substituting x = 0 into f(x) and g(x).
4. Step 4: Find f'(0) and g'(0) by differentiating f(x) and g(x) and then substituting x = 0 into the derivatives.
5. Step 5: Substitute f(0), g(0), f'(0), and g'(0) into the product rule expression for p'(0) to find the value of p'(0).

Key Concepts

Product Rule

The Product Rule is a fundamental principle in calculus used to differentiate the product of two functions. It states that if you have two functions, u(x) and v(x), the derivative of their product is given by p'(x) = u'(x)v(x) + u(x)v'(x). This rule is essential for finding the derivative of p(x) without directly computing p'(x).

Evaluating Derivatives at a Point

To find p′(0), we need to evaluate the derivative of p(x) at x = 0. This involves substituting x = 0 into the derivative expression obtained from the Product Rule. Understanding how to evaluate functions and their derivatives at specific points is crucial for solving problems that require finding instantaneous rates of change.

Exponential and Polynomial Functions

In the given function p(x), we have both exponential (e^x) and polynomial terms (like 10x⁵ and 20x³). Knowing how to differentiate these types of functions is vital, as they behave differently under differentiation. Exponential functions grow rapidly, while polynomial functions have a more predictable growth pattern, which influences the overall behavior of p(x) and its derivative.