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Understanding the concept of a derivative is key to solving problems involving tangent lines. A derivative represents the rate at which a function is changing at any given point. It is essentially the "slope" of the function at a particular location on its curve. When you need to find the equation of a tangent line to a curve, the derivative gives you the slope of that tangent line.

In our case, the function is given as \(y = x \sqrt{x}\). By rewriting this as \(y = x^{3/2}\), we can more easily differentiate it. Using the derivative, we can find out how the curve behaves at different points. Specifically, we need the slope of the tangent line that serves as the instantaneous rate of change of the function at any point \(x\).

The derivative of the function \(y = x^{3/2}\) is found using differentiation rules. Through this derivative, \(\frac{dy}{dx} = \frac{3}{2}x^{1/2}\), we can ascertain the slope of the tangent line we are seeking.

The power rule is a fundamental technique used in calculus to find the derivative of a function that is a power of \(x\). In simple terms, if you have a function \(x^n\), the derivative with respect to \(x\) is \(nx^{n-1}\).

Using the power rule for our function \(y = x^{3/2}\), we take the exponent \(3/2\), multiply it by \(x\), and reduce the exponent by 1. Thus:

• Original Function: \(x^{3/2}\)
• Derivative: \(\frac{3}{2}x^{1/2}\)

The power rule simplifies the process of finding derivatives for polynomial functions or functions that can be rewritten in polynomial form. It is a reliable method to quickly find the slope for these kinds of calculations.

The slope of a line represents how steep the line is. It's calculated as the "rise over run," or how much the line goes up (or down) for a given distance along the horizontal axis. In the context of tangent lines, the slope is critical because the slope of the tangent line must match the rate of change of the curve at that point.

For a line expressed as \(y = mx + b\), \(m\) is the slope. In our example, the slope of the given line \(y = 1 + 3x\) is 3. This means any tangent line that is parallel to this line will also have a slope of 3.

When we differentiate the curve, we set this derivative equal to the slope of the line we are interested in because we want the two lines to be parallel. This is how we determine the specific point on the curve \(y = x \sqrt{x}\) where the tangent line has this same slope.

Point-slope form is a method used to find the equation of a line when you know both a point on the line and its slope. The formula is:

• \(y - y_1 = m(x - x_1)\)

Where \((x_1, y_1)\) are the coordinates of the known point, and \(m\) is the slope of the line. This form is especially useful for quickly deriving the equation of a tangent line once you have all the necessary components.

In the exercise, after finding the point of tangency as \((4, 8)\) and knowing the slope is 3, we applied the point-slope form:

• Setup: \(y - 8 = 3(x - 4)\)

Expanding and simplifying gives us the equation of the tangent line: \(y = 3x - 4\). This derivation is straightforward with point-slope form, ensuring you're quickly able to write the equation as needed.