Derivatives: Complete Lesson with Animated Tangent and Key Formulas

The idea in one sentence

A derivative measures a function’s local slope — how $f$ changes “just next to” a point.

Equivalently: it’s the slope of the tangent line to the curve at that point.

See it: the tangent in action

Slide the cursor to move the point along the curve. The red tangent follows, and its slope updates live:

x₀ =1

Point

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Pente de la tangente (nombre dérivé)

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Things to notice:

• When $x_0 < 0$ → slope is negative (curve going down)
• When $x_0 = 0$ → slope is zero (bottom of parabola)
• When $x_0 > 0$ → slope is positive (curve going up)

Formula check: $f'(x) = 2x$. At $x_0 = 1$, slope = 2. At $x_0 = -2$, slope = $-4$.

Rigorous definition

The derivative of $f$ at $a$, written $f'(a)$, is the limit (if it exists):

$f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

The quotient $\dfrac{f(a + h) - f(a)}{h}$ is the difference quotient between $a$ and $a+h$: the slope of the secant line through those two points. As $h \to 0$, the secant tends to the tangent.

Equation of the tangent line

The tangent to $f$ at $x = a$ has equation:

$y = f'(a) \cdot (x - a) + f(a)$

Example: For $f(x) = x^2$ at $a = 3$:

• $f(3) = 9$
• $f'(3) = 6$
• Tangent: $y = 6(x - 3) + 9 = 6x - 9$.

Standard derivative formulas

Memorize these. Everything builds on them.

$f(x)$	$f'(x)$	Domain
$k$ (constant)	$0$	$\mathbb{R}$
$x$	$1$	$\mathbb{R}$
$x^n$ ($n \in \mathbb{Z}$)	$n \cdot x^{n-1}$	$\mathbb{R}$ (or $\mathbb{R}^*$ if $n < 0$)
$\sqrt{x}$	$\dfrac{1}{2\sqrt{x}}$	$x > 0$
$\dfrac{1}{x}$	$-\dfrac{1}{x^2}$	$x \neq 0$
$\ln(x)$	$\dfrac{1}{x}$	$x > 0$
$e^x$	$e^x$	$\mathbb{R}$
$\sin(x)$	$\cos(x)$	$\mathbb{R}$
$\cos(x)$	$-\sin(x)$	$\mathbb{R}$

Derivative operations

Operation	Formula
Sum	$(u + v)' = u' + v'$
Constant × function	$(k \cdot u)' = k \cdot u'$
Product	$(uv)' = u'v + uv'$
Quotient	$\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$
Chain rule	$(v \circ u)' = u' \cdot (v' \circ u)$

Examples

Example 1 (sum): $f(x) = 3x^2 - 5x + 7$

$f'(x) = 6x - 5$.

Example 2 (product): $f(x) = x^2 \cdot e^x$

$f'(x) = 2x \cdot e^x + x^2 \cdot e^x = e^x(2x + x^2)$.

Example 3 (quotient): $f(x) = \dfrac{x}{x^2 + 1}$

$f'(x) = \dfrac{1 \cdot (x^2 + 1) - x \cdot 2x}{(x^2 + 1)^2} = \dfrac{1 - x^2}{(x^2 + 1)^2}$.

Example 4 (chain): $f(x) = \sin(3x + 1)$

$f'(x) = 3 \cdot \cos(3x + 1)$.

Key applications

1. Monotonicity

$f' > 0$ on an interval ⇒ $f$ is increasing. $f' < 0$ ⇒ decreasing. $f' = 0$ at an isolated point ⇒ local extremum (confirm with sign table).

2. Finding an extremum

$f(x) = -x^2 + 4x + 1$.

• $f'(x) = -2x + 4$.
• $f' = 0 \iff x = 2$.
• Sign change (+ then −) → max at 2, value $f(2) = 5$.

3. Tangent equation

See section above.

Common mistakes

1. Forgetting $u'$ in a chain rule: $(\sin(2x))' = 2\cos(2x)$, NOT $\cos(2x)$.
2. Confusing product and composition: $(uv)' \neq (v(u))'$.
3. Sign in quotient rule: it’s $u'v - uv'$ (minus).
4. Derivative of a constant is 0, not the constant.

Solved exercises

Exercise 1

Compute the derivative of $f(x) = (x^2 + 1)^3$.

Show solution

Chain rule: $u = x^2 + 1$, $u' = 2x$. Outer: $v(x) = x^3$, $v'(x) = 3x^2$. $f'(x) = 2x \cdot 3(x^2 + 1)^2 = 6x(x^2 + 1)^2$.

Exercise 2

Find the tangent to $f(x) = e^x$ at $x = 0$.

Show solution

$f(0) = 1$, $f'(0) = 1$. Tangent: $y = x + 1$.

Exercise 3

Study the monotonicity of $f(x) = x^3 - 3x + 2$ on $\mathbb{R}$.

Show solution

$f'(x) = 3(x-1)(x+1)$. Positive for $x < -1$, negative on $(-1, 1)$, positive for $x > 1$. So $f$ is increasing on $(-\infty, -1]$, decreasing on $[-1, 1]$, increasing on $[1, +\infty)$. Local max at $-1$: $f(-1) = 4$. Local min at $1$: $f(1) = 0$.

Takeaways

• $f'(a)$ = slope of the tangent at $a$.
• Tangent: $y = f'(a)(x - a) + f(a)$.
• Monotonicity = sign of the derivative.
• Local extremum ⇒ $f' = 0$ (converse not always).
• Master the 9 standard formulas and 5 operations.

Going further

• Related tips: Essential derivative formulas, Chain rule, Product rule.
• Next: antiderivatives / integrals (reverse operation).
• Applications: optimization, kinematics (velocity = derivative of position).