The Derivative

The Tangent Line Equation

Use the derivative to write the equation of the tangent line to f(x)=x² at x=1.

tangent-line The tangent line to f at x=a has slope f'(a) and passes through (a, f(a)). Its equation is y = f(a) + f'(a)(x-a).

Example

Use the derivative slope and function point to write a tangent line.

highlighted = computed this step

Step 1 — Set up

Use the point where x equals 1.

f(x)=x2a=1f(x)= x^{2} \quad a= 1

Step 2 — Find the slope

Evaluate f prime at 1 to get slope 2.

m=f′(1)=2m=f'( 1 )= \hl{2}

Step 3 — Find the point

Evaluate f at 1 to get the point.

(1,1)( 1 , \hl{1} )

Step 4 — Write point-slope form

Write the tangent line in point-slope form.

y−1=2(x−1)y- \hl{1} = \hl{2} (x- 1 )

Step 5 — Result

State the tangent line.

y=1+2(x−1)y= \hlmath{1+2(x-1)}