Tangent Line

Precise definition

For a differentiable function $f$, the tangent line at $x=a$ is $L(x)=f(a)+f'(a)(x-a)$. Its slope is the derivative $f'(a)$, defined by the limit of secant slopes when that limit exists. The tangent is the best first-order local linear approximation.

Notation and mathematical language

Point–slope form $y-y_0=m(x-x_0)$ keeps the contact point visible. A vertical tangent may occur when slopes become unbounded, but then $f'(a)$ is not a finite real number under the standard derivative definition.

Conceptual picture

As $x$ approaches $a$, the difference $f(x)-L(x)$ is small relative to $x-a$: differentiability means $f(x)=L(x)+o(|x-a|)$. A tangent can cross the graph and need not touch it only once.

Conditions and key results

Continuity alone does not guarantee a tangent slope, as $|x|$ at zero shows. The derivative rule must match the function and domain. Linear approximation is local; error generally grows farther from $a$.

A reliable strategy

1. Confirm $a$ lies in the domain and differentiability is justified.
2. Compute the point $(a,f(a))$ and exact derivative $f'(a)$.
3. Substitute into $L(x)=f(a)+f'(a)(x-a)$ and simplify only after the structure is correct.
4. Verify the point and slope, then assess approximation near—not arbitrarily far from—$a$.

Fully worked example

Interpretation and application

Tangent lines approximate measurement transformations, marginal cost, motion, and nonlinear models. An exact local derivative of a fitted model does not eliminate data and model uncertainty.

Common mistakes

Verification and reasonableness

• Substitute $x=a$ and recover $f(a)$.
• Differentiate the line and recover slope $f'(a)$.
• Compare with secant slopes or nearby numerical values.

Practice

1. Find tangent slope of $x^2$ at $x=3$.
2. Find tangent line there.
3. Does continuity ensure differentiability?

Further deduction

Taylor's theorem refines the approximation. If $f''$ is continuous near $a$, then $f(x)=L(x)+\tfrac12f''(\xi)(x-a)^2$ for some intermediate $\xi$. A bound on $|f''|$ therefore gives a quantitative tangent-line error bound proportional to squared distance.

A tangent model is local, not automatically global. At a differentiable point $a$, $L(x)=f(a)+f'(a)(x-a)$ approximates $f(x)$ with error small relative to $|x-a|$ as $x\to a$. Tangents need not merely 'touch' a curve: $y=x^3$ crosses its horizontal tangent at the origin. At a corner such as $|x|$ at zero, unequal one-sided slopes mean no ordinary tangent line; at a vertical tangent, the derivative as a finite number fails even though a geometric tangent may be vertical. These cases show why the derivative limit, rather than a visual touching rule, is the definition.

Units pass through the tangent model: if $x$ is seconds and $f$ metres, then $f'(a)$ is metres per second and $f'(a)(x-a)$ is metres. Adding it to $f(a)$ is dimensionally valid, unlike adding a bare slope to a position.

Related topics

• Rate of Change
• Derivative
• Calculus Readiness