How do you find the slope of the tangent line to the parabola described by the equation y = 8x – x^2 at the point (1, 7)?

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To find the slope of the tangent line to the parabola given by the equation y = 8x – x2 at the point (1, 7), we need to perform the following steps:

Step 1: Differentiate the Equation

The first step is to calculate the derivative of the function with respect to x. The derivative will give us the slope of the tangent line at any point on the curve. We calculate:

y = 8x - x2

Using basic differentiation rules:


\frac{dy}{dx} = \frac{d}{dx}(8x) - \frac{d}{dx}(x2)  
= 8 - 2x

So, we find:


\frac{dy}{dx} = 8 - 2x

Step 2: Evaluate the Derivative at x = 1

Now that we have the derivative, we can substitute x = 1 to find the slope of the tangent line at the point (1, 7).


\frac{dy}{dx} \bigg|_{x=1} = 8 - 2(1)  
= 8 - 2  
= 6

Step 3: Conclusion

The slope of the tangent line to the parabola at the point (1, 7) is 6. Therefore, the tangent line at this point has a slope of 6.

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