Tangent Lines

Find the tangent line to f($x$) = $x^2$ at $x=2$.

• 1Find the point: f($2$) = $2^2=4$, so the point is ($2$, $4$)
• 2Find the gradient: f'($x$) = $2x$, so f'($2$) = $4$
• 3Substitute into $y$ − f(a) = f'(a)($x$ − a): $y-4=4$($x-2$)
• 4Expand and simplify: $y-4=4x-8$, so $y=4x-4$

Answer: $y=4x-4$

Find the tangent line to f($x$) = $x^3$ at $x=1$.

• 1Find the point: f($1$) = $1^3=1$, so the point is ($1$, $1$)
• 2Find the gradient: f'($x$) = $3x^2$, so f'($1$) = $3$
• 3Substitute: $y-1=3$($x-1$)
• 4Expand: $y-1=3x-3$, so $y=3x-2$

Answer: $y=3x-2$