Basic Properties and Formulas

These rules assume the component functions are differentiable at the evaluation points. A quotient or reciprocal additionally requires a nonzero denominator. For $g\circ h$, require $h$ differentiable at $x$ and $g$ differentiable at $h(x)$, with the composition defined nearby.

In the power rule, $n$ is a constant, not a function of $x$. For arbitrary real $n$, use $x>0$; positive integer powers extend to all real $x$, and negative integer powers exclude zero. At boundaries such as $x=0$ for a fractional power, check the definition separately. See the derivation and examples of these formulas.

Basic Derivative Rules

• Constant Rule: The derivative of a constant is zero.

  $$\frac{d}{dx} c = 0$$

• Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant times the derivative of the function.

  $$\frac{d}{dx} [c f(x)] = c f'(x)$$

• Power Rule: The derivative of a variable raised to a power is the power multiplied by the variable raised to one less than the power.

  $$\frac{d}{dx} x^n = nx^{n-1}$$

Derivative of Composite Functions

Sum Rule

The derivative of a sum of two functions is the sum of their derivatives.

$$\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$$

Difference Rule

The derivative of a difference between two functions is the difference of their derivatives.

$$\frac{d}{dx} [f(x) - g(x)] = f'(x) - g'(x)$$

Product Rule

The derivative of the product of two functions is given by:

$$\frac{d}{dx} [f(x) g(x)] = f'(x) g(x) + f(x) g'(x)$$

Quotient Rule

The derivative of the quotient of two functions is:

$$\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$

Chain Rule

Formal Definition

For two functions $g$ and $h$, with $y = g(u)$ and $u = h(x)$, the derivative of $y$ with respect to $x$ is:

$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

Lagrange Notation

Using Lagrange's notation, where $'$ denotes the derivative, for $y = g(h(x))$, the chain rule is expressed as:

$$y'(x) = g'(h(x)) \cdot h'(x)$$

Generalization for Multiple Compositions

For compositions involving multiple functions, such as $F = f \circ g \circ h$, the chain rule extends as follows: Here $u=h(t)$, $v=g(u)$, $y=f(v)$, and $F(t)=y$.

$$\frac{dF}{dt} = \frac{dy}{dv} \cdot \frac{dv}{du} \cdot \frac{du}{dt}$$

Or, using Lagrange notation for a function $F(t)$ that is a composition of $f$, $g$, and $h$, we get:

$$(f \circ g \circ h)'(t) = f'(g(h(t))) \cdot g'(h(t)) \cdot h'(t)$$

For a geometric view of composition, follow the three number lines in 3Blue1Brown’s chain-rule explanation: input, inner output, then outer output. The two successive changes show why the local scale factors multiply.

Special Derivatives

• Derivative of a Square: The derivative of the square of a function is twice the function times the derivative of the function.

  $$\frac{d}{dx} [f(x)]^2 = 2 f(x) f'(x)$$

• Derivative of a Reciprocal: The derivative of the reciprocal of a function.

  $$\frac{d}{dx} \left( \frac{1}{g(x)} \right) = -\frac{g'(x)}{[g(x)]^2}$$

Why the product rule has two terms

For $P=fg$, add and subtract $f(x)g(x+h)$ in the numerator:

$$\frac{P(x+h)-P(x)}h =\frac{f(x+h)-f(x)}h\,g(x+h) +f(x)\frac{g(x+h)-g(x)}h.$$

Taking limits gives $P'=f'g+fg'$ because differentiability of $g$ implies continuity. It is not $f'g'$: taking $f(x)=g(x)=x$ would incorrectly give $1$ instead of $2x$.

For a worked combination, let $y=x^2\sin(3x)$. The outer operation is a product; inside its second factor is a composition. Hence

$$y'=2x\sin(3x)+x^2\cos(3x)\cdot3.$$

The factor $3$ is the derivative of the inner input $3x$. The chain rule composes local rates; the Leibniz notation is a mnemonic, not a justification for cancelling arbitrary symbols.

Open full-size image

For positive side lengths and increments, the two strips contribute f Δg and g Δf; the corner contributes Δf Δg. Divide the area change by Δx. For differentiable f and g, the corner term tends to zero, leaving the two terms in the product rule. The limit proof above also covers signed values and decreases.

Peter Keep, Discover Calculus, Figure 2.4.4. · CC BY 4.0

References

• tutorial.math.lamar.edu/pdf/calculus_cheat_sheet_derivatives.pdf
• personal.math.ubc.ca/~CLP/CLP1/clp_1_dc_text.pdf
• mitres_18_001_f17_full_book.pdf

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