Derivatives from graphs Use the figure to find the following derivatives. <IMAGE>
d/dx (f(x)g(x)) | x=4

Guida verificata passo dopo passo

Step 1: Recall the product rule for derivatives, which states that if you have two functions f(x) and g(x), the derivative of their product is given by \( \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x) \).

Step 2: Identify the values of f(x), g(x), f'(x), and g'(x) at x = 4 from the graph. You will need to find the y-values of f(x) and g(x) at x = 4, as well as the slopes of the tangent lines to f(x) and g(x) at x = 4.

Step 3: Substitute the values of f(x), g(x), f'(x), and g'(x) at x = 4 into the product rule formula. This will give you the expression for the derivative of the product at x = 4.

Step 4: Simplify the expression obtained in Step 3 to find the derivative of the product at x = 4.

Step 5: Verify your result by checking the calculations and ensuring that the values from the graph are correctly interpreted.

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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, f(x) and g(x), the derivative of their product is given by f'(x)g(x) + f(x)g'(x). This rule is essential for finding the derivative of the expression d/dx (f(x)g(x)) at a specific point, such as x=4.

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The Product Rule

Evaluating Derivatives at a Point

Evaluating derivatives at a specific point involves substituting the value of x into the derivative function obtained from differentiation. In this case, after applying the Product Rule, you will need to calculate the values of f(4), g(4), f'(4), and g'(4) to find the derivative at x=4. This step is crucial for obtaining a numerical answer.

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Critical Points

Graphical Interpretation of Derivatives

The graphical interpretation of derivatives involves understanding how the slope of the tangent line to a curve at a given point represents the derivative at that point. By analyzing the graph of f(x) and g(x), one can visually assess the behavior of the functions and their derivatives, which aids in comprehending the results obtained through algebraic differentiation.

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