Slopes and Tangent Lines

In Exercises 13–16, differentiate the functions and find the slope of the tangent line at the given value of the independent variable.

f(x) = x + 9/x, x = −3

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First, identify the function you need to differentiate: \( f(x) = x + \frac{9}{x} \). This function is composed of two terms: a linear term \( x \) and a rational term \( \frac{9}{x} \).

Differentiate each term separately. The derivative of \( x \) with respect to \( x \) is 1. For the term \( \frac{9}{x} \), rewrite it as \( 9x^{-1} \) and use the power rule to differentiate, which gives \( -9x^{-2} \).

Combine the derivatives of the individual terms to find the derivative of the entire function: \( f'(x) = 1 - \frac{9}{x^2} \).

Substitute the given value of the independent variable \( x = -3 \) into the derivative to find the slope of the tangent line at that point: \( f'(-3) = 1 - \frac{9}{(-3)^2} \).

Simplify the expression to find the slope of the tangent line at \( x = -3 \). This involves calculating \( (-3)^2 \) and simplifying the fraction \( \frac{9}{9} \).

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주요 개념

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Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its independent variable. For the function f(x) = x + 9/x, differentiation involves applying the power rule and the quotient rule to find f'(x), the derivative of f(x).

Slope of the Tangent Line

The slope of the tangent line to a curve at a given point is the value of the derivative at that point. It represents the instantaneous rate of change of the function. For f(x) = x + 9/x at x = -3, the slope is found by evaluating the derivative f'(x) at x = -3.

Evaluating Derivatives at a Point

Once the derivative of a function is determined, it can be evaluated at a specific point to find the slope of the tangent line at that point. This involves substituting the given value of the independent variable into the derivative. For f(x) = x + 9/x, substitute x = -3 into f'(x) to find the slope.

Slopes and Tangent LinesIn Exercises 13–16, differentiate the functions and find the slope of the tangent line at the given value of the independent variable.f(x) = x + 9/x, x = −3