3.3 – Rules for Differentiation

Constants

The derivative of a constant is zero:

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

Constant multiples can be brought outside the function:

$$\frac{d}{dx}cu=c\frac{du}{dx}=c{u}^{\prime }$$

Sums and Differences

A function that is a sum or difference of two functions can be broken up:

$$\frac{d}{dx}u\pm v=\frac{du}{dx}\pm \frac{dv}{dx}=u^{\prime }\pm v^{\prime }$$

Powers

The power rule is as follows:

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

Products

The product rule is as follows:

$$\frac{d}{dx}uv=u^{\prime }v+u{v}^{\prime }$$

Quotients

The quotient rule is as follows:

$$\frac{d}{dx}\frac{u}{v}=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$$

where $v\ne 0$.

Examples

Differentiate $2x^2+3x-5$

Solution

Apply the sum and difference rules

$$\frac{d}{dx}2x^2+3x-5=\frac{d}{dx}2x^2+\frac{d}{dx}3x-\frac{d}{dx}5$$

Apply the power rule, constant rule

$$=2\cdot 2x^{2-1}+3x^{1-1}-0$$ $$=4x+3$$

Differentiate $\frac{12x^3+x}{3x+2}$

Solution

Let $u=12x^3+x$, $v=3x+2$

Apply the power rule and constant rule to find $u^{\prime }=36x^2+1$, $v^{\prime }=3$

Substitute into the quotient rule

$$\frac{d}{dx}\frac{12x^3+x}{3x+2}=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$$ $$=\frac{(36x^2+1)(3x+2)+(12x^3+x)(3)}{(3x+2{)}^2}$$

Expand and collect like terms if possible

$$=\frac{72x^3+72x^2+2}{(3x+2{)}^2}$$

Find the tangent line of $f(x)=4x^3+2x+3$ at $x=2$

Solution

Differentiate $f(x)$

$$f^{\prime }(x)=3\cdot 4x^{3-1}+2x^{1-1}+0$$ $$12x^2+2$$

Find the derivative at $x=2$

$$f^{\prime }(2)=12(2{)}^2+2=50$$

Substitute into the point-slope form at $(2,f(2))=(2,39)$

$$y=50(x-2)+39$$

Note that values of $x$ in the neighbourhood of $a$ for the tangent line at $a$ are approximately equal to $f(a)$. This is called the local linearization.

Higher-Order Derivatives

The second derivative is shown as:

$$\frac{d^{2}y}{d{x}^2}\text{ or }{y}^{\prime \prime }$$

And for higher-order derivatives:

$$\frac{d^{n}y}{d{x}^n}\text{ or }{y}^{(n)}$$

Applications

	Roots	$(+)$	$(-)$
$\frac{d}{dx}$	Minima\Maxima	Increasing	Decreasing
$\frac{d^2}{d{x}^2}$	Inflection points	Concave up	Concave down

Example

Find the velocity and acceleration of a particle with displacement given by the function $d=12t^2+4t+2$

Velocity

Velocity $v$ is the first derivative of displacement $d^{\prime }$

$$d^{\prime }=24t+4$$ $$v=24t\text{ m/s}$$

Acceleration

Acceleration $a$ is the first derivative of velocity $v^{\prime }$ or alternatively, the second derivative of displacement $d^{\prime \prime }$

$$d^{\prime \prime }=24$$ $$a=24{\text{ m/s}}^2$$