A tangent line is a straight line that touches a curve at a single point. The tangent slope is the slope of the tangent line. Make sure your curve and tangent line are drawn on a graph with grid lines. This will make it easier to calculate the tangent slope. Since this is a hand-drawn method, this calculation will only be an estimate, not the exact derivative at a point.
slope = (y2 - y1) / (x2 - x1) using the points (x1, y1) and (x2, y2) for example, if you have the points (1, 3) and (3, 7), slope = (7 - 3) / (3 - 1) slope = 4 / 2 slope = 2
(df / dx)(x) = [f(x + dx) – f(x)] / dx which can be written as f’(x) = [f(x + dx) – f(x)] / dx where f(x) is the function f of x (sometimes written as “y”), i. e. how the value of y changes as the value of x changes f’(x) is the derivative of f(x), as indicated by the prime symbol (’) dx is a small change in x that approaches 0 f(x + dx) is the value of y at the horizontal value x + dx
For more general calculus tips, see our guide on how to do well in calculus. If you’re looking for how to use derivative rules, check out How to Differentiate Polynomials.
f’(x) = [(x + dx)^2 – (x)^2] / dx
f’(x) = [(x + dx)^2 – (x)^2] / dx
f’(x) = [(x + dx)^2 – (x)^2] / dx
Our starting equation: f’(x) = [(x + dx)^2 – (x)^2] / dx Writing out the expanded polynomial term: f’(x) = [x^2 + 2xdx + dx^2 – x^2] / dx The terms x^2 and – x^2 equal zero, resulting in: f’(x) = [2xdx + dx^2] / dx Both terms in the numerator have a dx, which can cancel out with the dx in the denominator, giving the simplified equation: f’(x) = 2x + dx
f’(x) = 2x + 0 which simplifies to: f’(x) = 2x So, the derivative of f(x) = x^2 is f’(x) = 2x
Find the slope of the tangent line at x = 5 for the function f(x) = x^2. f’(x) = 2x f’(5) = 2(5) f’(5) = 10 The slope of the tangent line at x = 5 is 10.
