Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.17

Chapter 3, Problem 3.17

The line tangent to the graph of f at x=5 is y = 1/10x-2. Find d/dx (4f(x)) |x+5

Verified step by step guidance

Step 1: Understand the problem. We need to find the derivative of the function 4f(x) at x = 5. The tangent line equation y = \(\frac{1}{10}\)x - 2 gives us information about the derivative of f(x) at x = 5.

Step 2: Recall that the slope of the tangent line to the graph of f at x = 5 is the derivative of f at that point, f'(5). From the equation y = \(\frac{1}{10}\)x - 2, the slope is \(\frac{1}{10}\). Therefore, f'(5) = \(\frac{1}{10}\).

Step 3: Use the constant multiple rule for derivatives. The derivative of 4f(x) with respect to x is 4f'(x).

Step 4: Substitute x = 5 into the derivative expression. We have 4f'(5).

Step 5: Substitute the value of f'(5) from Step 2 into the expression from Step 4. This gives us 4 \(\times\) \(\frac{1}{10}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which is given by the derivative. In this case, the equation of the tangent line provides the slope and y-intercept needed to understand the behavior of the function f near x=5.

Derivative

The derivative of a function measures how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In this problem, we need to find the derivative of 4f(x) at x = -5, which involves applying the rules of differentiation to the function f.

Chain Rule

The chain rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. In this context, understanding how to apply the chain rule will be essential for differentiating 4f(x) effectively.

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