CalcNotes Differentiation Rules

Section 3.3 Differentiation Rules

Theorem 3.3.1. Derivative of constant function.

\(\frac{d}{dx} c=0\) for any constant \(c\text{.}\)

Proof.

For \(f(x)=c\text{,}\)

\begin{equation*} f'(x)=\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h} = \lim_{h\rightarrow 0} \frac{c-c}{h} = \lim_{h\rightarrow 0} \frac{0}{h} = \lim_{h\rightarrow 0} 0 = 0 \end{equation*}

Fact 3.3.2. Power rule.

\(\frac{d}{dx} x^n = nx^{n-1}\) for any real number \(n\) and over all \(x\) for which both \(x^n\) and \(x^{n-1}\) are defined.

Example 3.3.3.

Compute the derivatives of:

\(\displaystyle f(x)=x^5\)
\(\displaystyle y=\frac{1}{x^3}\)
\(\displaystyle g(x) = \sqrt{x}\)

Answer.

\(\displaystyle f'(x) = 5x^4\)
\(\displaystyle \frac{dy}{dx} = \frac{-3}{x^4}\)
\(\displaystyle g'(x) = \frac{1}{2\sqrt{x}}\)

Fact 3.3.4. Rules for constant multiples and sums.

If \(u\) and \(v\) are differentiable functions of \(x\) (i.e., \(u=f(x)\) and \(v=g(x)\)), and \(c\) is a constant, then:

\(\displaystyle \frac{d}{dx} cu = c \frac{du}{dx}\)
\(\displaystyle \frac{d}{dx} u + v = \frac{du}{dx} + \frac{dv}{dx}\)

Example 3.3.5.

Compute \(\frac{dy}{dx}\) for \(y=3x^{\sqrt{2}} - 6 \sqrt[3]{x}\text{.}\)

Answer.

\(\frac{dy}{dx} = 3\sqrt{2}x^{\sqrt{2}-1} - 2 x^{-2/3}\)

Example 3.3.6.

Let \(f(x)=x^3-9x+2\text{.}\)

Find \(f'(x)\text{.}\)
Find the equation of the tangent line at \(x=2\text{.}\)
Find any points where the tangent line is horizontal.

Answer.

\(\displaystyle f'(x) = 3x^2 - 9\)
\(\displaystyle y=3x-14\)
\(x=\sqrt{3}\) and \(-\sqrt{3}\)

Fact 3.3.7. Derivative of \(e^x\).

\begin{equation*} \frac{d}{dx}e^x=e^x \end{equation*}

Theorem 3.3.8. Product Rule.

If \(u\) and \(v\) are differentiable at \(x\text{,}\) then so is \(uv\) and

\begin{equation*} \frac{d}{dx} uv = u \frac{dv}{dx} + v\frac{du}{dx} \end{equation*}

In function notation,

\begin{equation*} \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x) \end{equation*}

\begin{align*} \lim_{h\rightarrow 0} \frac{f(x+h)g(x+h)-f(x)g(x)}{h} \amp = \lim_{h\rightarrow 0} \frac{f(x+h)g(x+h) - f(x+h)g(x)+f(x+h)g(x)-f(x)g(x)}{h}\\ \amp = \lim_{h\rightarrow 0} \left(\frac{f(x+h)g(x+h) - f(x+h)g(x)}{h} + \frac{f(x+h)g(x)-f(x)g(x)}{h}\right)\\ \amp = \lim_{h\rightarrow 0} \frac{f(x+h)g(x+h) - f(x+h)g(x)}{h} + \lim_{h\rightarrow 0} \frac{f(x+h)g(x)-f(x)g(x)}{h}\\ \amp = \lim_{h\rightarrow 0} f(x+h)\frac{g(x+h) - g(x)}{h} + \lim_{h\rightarrow 0} g(x)\frac{f(x+h)-f(x)}{h}\\ \amp = \left(\lim_{h\rightarrow 0} f(x+h)\right) \left(\lim_{h\rightarrow 0} \frac{g(x+h) - g(x)}{h}\right) + g(x)\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}\\ \amp = f(x) g'(x) + g(x)f'(x) \end{align*}

Example 3.3.9.

Compute the derivatives:

\(\displaystyle y=x^2e^x\)
\(\displaystyle f(x) = (x^2+3x+1)(x+1)\)

Answer.

\(\displaystyle \frac{dy}{dx} = 2xe^x + x^2e^x\)
\(\displaystyle f'(x) = (2x+3)(x+1) + (x^2+3x+1) = 3x^2 +8x+4\)

Fact 3.3.10. Quotient rule.

If \(u\) and \(v\) are differentiable at \(x\) and \(v(x)\neq 0\text{,}\) then \(\frac{u}{v}\) is differentiable at \(x\) and

\begin{equation*} \frac{d}{dx} \frac{u}{v} = \frac{v \frac{du}{dx} - u\frac{dv}{dx}}{v^2} \end{equation*}

\begin{equation*} \frac{d}{dx} \frac{f(x)}{g(x)} = \frac{f(x)g'(x) - f'(x)g(x)}{g^2(x)} \end{equation*}

Example 3.3.11.

Compute the derivatives:

\(\displaystyle f(x) = \frac{3x+4}{5-6x}\)
\(\displaystyle y= \frac{x^3-4e^x}{\sqrt{x}+2}\)

Answer.

\(\displaystyle f'(x) = \frac{39}{(5-6x)^2}\)
\(\displaystyle \frac{dy}{dx} = \frac{(\sqrt{x}+2)(3x^2-4e^x) - (x^3-4e^x)(\frac{1}{2\sqrt{x}}}{(\sqrt{x}+2)^2}\)

Example 3.3.12.

Let \(\displaystyle u=\frac{5t^{4/3}-2e^3}{\sqrt{t}}\text{.}\) Find \(u'\text{.}\)
Let \(f(x)=7\sqrt[5]{x^3} + 2x^e\text{.}\) Find \(f'(x)\text{.}\)

Answer.

\(\displaystyle \displaystyle u' = \frac{25}{6}e^{-1/6} + e^3 t^{-3/2}\)
\(\displaystyle \displaystyle f'(x) = \frac{21}{5}x^{-2/5} + 2ex^{e-1}\)

Definition 3.3.13. Higher order derivatives.

The derivative of the derivative is called the second derivative and is denoted:

\begin{equation*} f''(x) = \frac{d^2 y}{dx^2} = \frac{d}{dx}\frac{dy}{dx} = y'' = D^2(f)(x) = D^2_x f(x) \end{equation*}

Similarly, the third derivative is:

\begin{equation*} f'''(x) = \frac{d^3 y}{dx^3} = y''' = D^3 f(x) \end{equation*}

For larger integers \(n\text{:}\)

\begin{equation*} f^{(n)} = y^{(n)} = \frac{d^n y}{dx^n} = D^n f \end{equation*}

The positive integer \(n\) is called the order.

Example 3.3.14.

Let \(f(x) = 3x^4+5x^2-6x+2\text{.}\) Find the derivatives of all orders.

Answer.

\begin{align*} f'(x) \amp = 12x^3 + 10x -6\\ f''(x) \amp = 36x^2 + 10\\ f^{(3)}(x) \amp = 72x\\ f^{(4)}(x) \amp = 72\\ f^{(5)}(x) \amp = 0 \end{align*}

Example 3.3.15.

Find \(\frac{d^2}{dx^2} \sqrt{x}\)

Answer.

\(\displaystyle\frac{-1}{4}x^{-3/2}\)

Example 3.3.16.

Let \(s=t^3 e^t\text{.}\) Find \(s'\) and \(s''\text{.}\)

Answer.

\begin{align*} s' \amp = 3t^2e^t + t^3e^t\\ s'' \amp = 6te^t + 6t^2e^2 + t^3e^t \end{align*}