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To find the slope of the tangent line at a specific point on a curve, we use the derivative. The derivative of a function measures how the function's value changes as its input changes. In simpler terms, it tells us the 'rate of change' or the 'slope' of the function at any given point. For example, if we have a curve described by the function \( y = f(x) \), the derivative \( f'(x) \) represents the slope of the tangent line to the curve at any point \( x \). Understanding this concept is crucial for problems involving tangents and slopes.

The power rule is a basic rule in calculus used to find the derivative of functions that are polynomials. It states that if you have a function of the form \( f(x) = x^n \), where \( n \) is a constant, the derivative is found by multiplying the exponent by the base raised to the power of one less than the original exponent. Mathematically, the rule is written as: \[ \frac{d}{dx}(x^n) = nx^{n-1} \] For a polynomial like \( y = x^3 + 3x - 8 \), we apply the power rule to each term separately:
- The derivative of \( x^3 \) is \( 3x^2 \).
- The derivative of \( 3x \) is 3.
- The derivative of a constant, like -8, is 0.
This gives us the overall derivative \( \frac{dy}{dx} = 3x^2 + 3 \).

Once the derivative is found, the next step is to evaluate it at the given point to find the slope of the tangent line. This process involves substituting the \( x \)-value of the point into the derivative. For the function \( y = x^3 + 3x - 8 \), we found the derivative to be \( \frac{dy}{dx} = 3x^2 + 3 \). To find the slope at the point (2,6), we substitute \( x = 2 \) into the derivative:
\[ \frac{dy}{dx} = 3(2)^2 + 3 \] Simplifying this, we get:
\[ \frac{dy}{dx} = 3(4) + 3 = 12 + 3 = 15 \] Thus, the slope of the tangent line to the curve at the point (2, 6) is 15.