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nicfv authored Jan 6, 2025
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The tangent line to a curve is a line that passes through a single local point on the curve, that also matches the same slope. That means, the derivative of the curve must match the slope of the tangent line.

Let's assume our curve is defined by the formula below and we want to find the tangent line at $x_{0}=1$.
Let's assume our curve is defined by the formula below and we want to find the tangent line at \(x_{0}=1\).

$$f(x) = \frac{1}{8}x^{2} - x - 4$$

We could plug in $f(x_{0})$ to obtain $y_{0}$, then compute the derivative $f'(x_{0})$ to obtain the slope, and then plug in our values into point-slope form.
We could plug in \(f(x_{0})\) to obtain \(y_{0}\), then compute the derivative \(f'(x_{0})\) to obtain the slope, and then plug in our values into point-slope form.

$$y-y_{0}=m(x-x_{0})$$

Finally, we can rewrite this formula in y-intercept form $y=mx+b$ where $b=y_{0}-mx_{0}$.
Finally, we can rewrite this formula in y-intercept form \(y=mx+b\) where \(b=y_{0}-mx_{0}\).

$$y=mx+(y_{0}-mx_{0})$$

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