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All my math notes, now in Markdown.

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3-3 Differentiation Rules

Learning Targets

You will be able to

Concepts / Definitions

Derivative of a Constant

If $f$ is the function with constant value $c$, then \(\frac{df}{dx} = \frac{d}{dx}(c) = 0\)

Constant times a Function Rule

Solving for the rule

If $u$ is a differentiable function of $x$ and $c$ is a constant, then for $y = cu$ $\frac{d}{dx}cu = ?$

Definition

If $u$ is a differentiable function of $x$ and $c$ is a constant, then \(\frac{d}{dx}cu = c\frac{du}{dx}\)

The Sum and Difference Rule

Solving for the rule

If $u$ and $v$ are differentiable functions of $x$, then their sum and difference are differentiable at every point where $u$ and $v$ are differentiable. At such points, $\frac{d}{dx}(u\pm v) = ?$

\(y' = \lim_{h\to 0} \frac{u(x+h)+v(x+h)-(ux+vx)}{h}\) \(y' = \lim_{h\to 0} \frac{u(x+h)-ux+v(x+h)-vx}{h}\) \(y' = \lim_{h\to 0} \frac{u(x+h)-ux}{h} + \lim_{h\to 0} \frac{v(x+h)-vx}{h}\) \(y' = u'(x) + v'(x)\)

Definition

If $u$ and $v$ are differentiable functions of $x$, then their sum and difference are differentiable at every point where $u$ and $v$ are differentiable. At such points, \(\frac{d}{dx} (u \pm v) = \frac{du}{dx} \pm \frac{dv}{dx}\)

Power Rule for Derivatives

\[f(x) = x^2\]
x -1 0 1 2 3
f’(x) -2 0 2 4 6
\[f'(x) = 2x\] \[g(x) = x^3\]
x -1   0   1   2   3
g’(x) 3   0   3   12   27
      V   V   V   V
      -3   3   9   15
        V   V   V  
        6   6   6  
\[g'(x) = 3x^2\]

Power Rule

Definition

If $f(x) = x^n$ is a power function, with non-zero real number $n$, then \(\frac{d}{dx}f(x) = nx^{n-1}\)

The Product Rule

Solving

\(f'(x) = \lim_{h\to 0} \frac{(x+h)^n - x^n}{h}\) \(f'(x) = \lim_{h\to 0} \frac{(_nC_0x^nh^0 + _nC_1x^{n-1}h^1 + _nC_2x^{n-2}h^2 + ... + _nC_{n-1}x^1h^{n-1} + _nC_nx^0h^n) - x^n}{h}\) \(f'(x) = \lim_{h\to 0} \frac{(x^n + nx^{n-1}h^1 + _nC_2x^{n-2}h^2 + ... + nx^1h^{n-1} + h^n) - x^n}{h}\) \(f'(x) = \lim_{h\to 0} \frac{h(nx^{n-1} + _nC_2x^{n-2}h^1 + ... + nx^1h^{n-2} + h^{n-1})}{h}\) \(f'(x) = \lim_{h\to 0} (nx^{n-1} + _nC_2x^{n-2}h^1 + ... + nx^1h^{n-2} + h^{n-1})\) \(\)

Definition

The product of two differentiable functions $u$ and $v$ is differentiable, and \(\frac{d}{dx}(uv) = \frac{du}{dx}v+u\frac{dv}{dx}\)

The Quotient Rule

Definition

At a point where $v \neq 0$, the quotient $y = \frac uv$ of two differentiable functions is differentiable, and \(\frac{d}{dx}(\frac uv) = \frac{\frac{du}{dx}v - u\frac{dv}{dx}}{v^2}\)

In words, derivative of the numerator times the denominator minus the numerator times derivative of the denominator, all divided by the denominator squared.

Slightly simplier, derivative of the top times the bottom
minus the top times derivative of the bottom,
all divided by the bottom squared.

A common catch phrase to remember this is “Low d high minus high d low all divided by bottom squared”.

Higher Order Derivatives

Second Derivative

\(y'' = \frac{d}{dx}y' = \frac{d}{dx} (\frac{dy}{dx}) = \frac{d^2y}{dx^2}\)

Third Derivative

\(y''' = \frac{d}{dx}y'' = \frac{d}{dx}(\frac{dy'}{dx}) = \frac{d^3y}{dx^3}\)

nth Derivative

\(y^{(n)} = \frac{d}{dx}y^{n-1} = \frac{d^ny}{dx^n}\)

Examples

Find an equation of the line tangent to the curve $y = \frac{x^2 - 1}{x^2 + 1}$ at $x = -2$

\(\frac{dy}{dx} = \frac{4x}{(x^2+1)^2}\) \(\frac{dy}{dx}\rvert_{x = -2} = \frac{4(-2)}{((-2)^2 + 1)^2}\) \(\frac{dy}{dx}\rvert_{x = -2} = -\frac{8}{25}\) \(y = \frac{(-2)^2-1}{(-2)^2+1}\) \(y = \frac 35\) \(z = -\frac{8}{25}(x+2) + \frac 35\)

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