Slope of x²−6x+5 at x = 4

Given the function f(x) = x² − 6x + 5, find its derivative at x = 4 and the slope of the tangent line at that point.

Step-by-step

1. f(x) = 1x² − 6x + 5
   First identify the quadratic coefficient, linear coefficient, and constant term of the function, paying attention to signs.
2. f′(x) = 2x − 6
   Power rule for derivatives: the derivative of x² is 2x, of x is 1, and of a constant is 0
3. k = f′(4) = 2 × 4 − 6 = 2
   The value of the derivative at a point equals the slope of the tangent line there; this is the geometric meaning of the derivative.
4. y₀ = f(4) = -3
   Substitute the x-coordinate of the tangency point back into the original function to get its y-coordinate.
5. y − (-3) = 2(x − 4) ⇒ y = 2x − 11
   Write the tangent line equation in point-slope form: slope comes from the derivative, point is the tangency point.

Why it works

The geometric meaning of the derivative is the slope of the tangent line, so to find the slope you only need to differentiate and substitute the x-coordinate; there is no need to write the tangent equation first. Differentiate a polynomial term by term: reduce the power of the quadratic term by one and multiply by its coefficient, turn the linear term into a constant, and eliminate the constant term. When substituting, be careful to carry both the negative sign of the coefficient and the negative sign of the x-coordinate.

Common mistakes

• Forgetting to multiply the quadratic term by its original coefficient when differentiating.
• Treating the constant term as something that remains after differentiation.
• Missing a negative sign during substitution, leading to an incorrect slope sign.

Related

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