Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 3 - Derivatives

Problema 94

Capitolo 3, Problema 94

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

Guida verificata passo dopo passo

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.

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).

Step 3: Evaluate f(0) and g(0) by substituting x = 0 into f(x) and g(x).

Step 4: Find f'(0) and g'(0) by differentiating f(x) and g(x) and then substituting x = 0 into the derivatives.

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).

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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).

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The Product Rule

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.

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Critical Points

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.

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Exponential Functions

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