Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problema 3.2.43a

Capitolo 3, Problema 3.2.43a

Use the definition of the derivative to determine d/dx(ax²+bx+c), where a, b, and c are constants.

Guida verificata passo dopo passo

Step 1: Recall the definition of the derivative. The derivative of a function f(x) at a point x is given by the limit: \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \).

Step 2: Identify the function f(x) = ax^2 + bx + c. We need to find f(x+h) by substituting x with (x+h) in the function: f(x+h) = a(x+h)^2 + b(x+h) + c.

Step 3: Expand f(x+h). Use the distributive property to expand: a(x+h)^2 = a(x^2 + 2xh + h^2), and b(x+h) = bx + bh. So, f(x+h) = ax^2 + 2axh + ah^2 + bx + bh + c.

Step 4: Substitute f(x) and f(x+h) into the derivative definition: \( f'(x) = \lim_{h \to 0} \frac{(ax^2 + 2axh + ah^2 + bx + bh + c) - (ax^2 + bx + c)}{h} \).

Step 5: Simplify the expression inside the limit. Cancel out the terms ax^2, bx, and c, and then divide each remaining term by h: \( f'(x) = \lim_{h \to 0} \frac{2axh + ah^2 + bh}{h} = \lim_{h \to 0} (2ax + ah + b) \). Finally, evaluate the limit as h approaches 0.

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Definition of the Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. Formally, it is defined as the limit of the average rate of change of the function as the interval approaches zero. This concept is foundational in calculus, as it provides a way to understand how functions behave locally.

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Definition of the Definite Integral

Power Rule

The power rule is a basic differentiation rule that states if f(x) = x^n, then f'(x) = n*x^(n-1), where n is a constant. This rule simplifies the process of finding derivatives for polynomial functions, making it easier to differentiate terms like ax², bx, and c in the given expression.

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Linearity of Derivatives

The linearity of derivatives refers to the property that the derivative of a sum of functions is the sum of their derivatives, and the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. This principle allows for the straightforward differentiation of polynomial expressions by treating each term independently.

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Linearization

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