Find the equation of a tangent line
Use the derivative to get the slope at a point, then build the line y = m(x − a) + f(a).
A tangent line touches the curve at one point and has the same slope as the curve there. The slope is the derivative.
Take f(x) = x² at x = 1. The point is (1, 1) and the slope is 2, so the line is y = 2(x − 1) + 1 = 2x − 1.
Classic CAS
Start on the main screen: Press home to open the Home menu (Calculator, Graphs, Settings, and more). Choose Calculator, or press esc to go back to the calculation page.
Type expand( then the point-slope form: the derivative at x = 1 times (x − 1), plus the y value 1. Press enter.
expand(d(x^2▶,x=1)×(x−1)+1)enter
You should see2·x-1
Classic Graphing 84
Start on the main screen: Press 2nd then mode (QUIT) to get back to the home screen. Pressing clear on a menu also backs out of it.
First get the slope. MATH 8 is nDeriv: type X², X, 1 and close the bracket.
math8X,T,θ,nx²,X,T,θ,n,1)enterNow find the intercept b = y − m·x. The slope is stored in Ans (2nd, then the (−) key). Type 1 − Ans × 1.
1−2ndANS(−)×1enter
You should seeThe slope is 2 and the intercept is -1, so the tangent line is y = 2x − 1.
Natural Color Graphing
Start on the main screen: Press MENU then 1 to open Run-Matrix, the main calculation screen. Press EXIT to back out of a bar or screen, and SHIFT then EXIT (QUIT) to return to the app start.
Open the graph screen, enter X² and press EXE, then draw it with F6.
MENU3X,θ,Tx²EXEF6Press F4 (SKETCH), then F2 (tangent). Type 1 for the point and press EXE twice. The line is drawn and its slope is displayed.
F4F21EXEEXE
You should seedy/dx=2
Tips
- The slope of a normal line (at right angles) is the negative reciprocal of the tangent slope: -1/m.
Related guides
- Find the slope of a function at a pointGet the derivative of a function at one x value as a decimal, using the built-in derivative function.
- Read the slope from a graphGraph a function and move to a point to see dy/dx, the slope of the curve there.
- Find the slope of a lineWork out rise over run between two points, such as (2, 1) and (6, 4).