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Understanding how to calculate the derivative of a function is vital when finding the equation of a tangent line. The derivative represents the slope or rate of change of the function at any given point. For instance, in our example with the function \( f(x) = x^2 + 2x \), to find the derivative, we apply the power rule. This rule states that the derivative of \( x^n \) is \( nx^{n-1} \). Hence, the derivative of our function is:\[ f'(x) = \frac{d}{dx}(x^2 + 2x) = 2x + 2 \]With this expression, we can easily find the slope of the tangent at any point on the curve by substituting \( x \) with the desired value. In this case, we substitute \( x = 3 \) to find the slope of the tangent at the specific point. Doing this, we get:\[ f'(3) = 2(3) + 2 = 8 \] This value, 8, is the slope of the tangent line at \( x = 3 \).

The point-slope form is a straightforward and useful way to write the equation of a line when you know a point on the line and the slope. The standard format of point-slope form is:\[ y - y_1 = m(x - x_1) \]Here, \((x_1, y_1)\) represents the coordinates of the point through which the line passes, and \( m \) is the slope. In our problem, we've already calculated the slope \( m = 8 \) and found the point on the curve at \( x = 3 \), which is \( (3, 15) \). Substituting these parameters into the point-slope formula gives:\[ y - 15 = 8(x - 3) \]By expanding and simplifying this equation, we obtain:\[ y = 8x - 9 \] This is the equation of the tangent line in slope-intercept form, where it describes a linear relationship between \( x \) and \( y \).

Graph verification serves as a useful tool for confirming the accuracy of our mathematical calculations. By graphing both the original function \( f(x) = x^2 + 2x \) and the tangent line \( y = 8x - 9 \), we can visually inspect their relationship. Using a graphing calculator, plot both equations. At \( x = 3 \), observe how the tangent touches the curve exactly at one point without crossing it. This observation supports that the line is genuinely tangent to the curve at this point.

• Ensure the parabola of the function \( f(x) \) is plotted correctly.
• The tangent line should visibly touch the curve at exactly one point, confirming it is indeed tangent.
• Cross-check the coordinates of the tangent with your calculations to validate accuracy.

The slope of a curve at any given point is determined by its derivative at that point. This concept differs from lines which have a constant slope. For a curve, the slope varies depending on the location along it. In our problem, the slope of the curve \( f(x) = x^2 + 2x \) at \( x = 3 \) is the value of the derivative \( f'(3) = 8 \).

• The slope provides information about how steep the curve is at a particular point.
• A positive slope, as in our example, indicates the curve ascends to the right.
• If the derivative is zero at a point, the curve would be horizontal at that point.

Understanding this concept helps to deduce how the curve and its tangent are oriented relative to each other at specific points. Being able to calculate and interpret this slope is essential for solving problems involving tangents, and more generally for understanding the behavior of differentiable functions.