# How do you find the slope and tangent line to the curve #y=6-x^2# at x=7?

Slope

equation:

To find the slope you need to derive your function and evaluate the derivative at

deriving:

at

To find the equation of the tangent line you need also the value of

So basically your tangent has solpe

Now use the relationship:

Graphicall (the red line is the tangent):

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To find the slope of the curve at a specific point, we need to find the derivative of the function. The derivative of y=6-x^2 is dy/dx = -2x.

To find the slope at x=7, we substitute x=7 into the derivative: dy/dx = -2(7) = -14.

Therefore, the slope of the curve at x=7 is -14.

To find the equation of the tangent line at x=7, we use the point-slope form of a line: y - y1 = m(x - x1), where (x1, y1) is the point on the curve.

Substituting x=7, y=6-7^2=-43 into the equation, we have: y - (-43) = -14(x - 7).

Simplifying, we get the equation of the tangent line: y = -14x + 105.

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