Differentiation

There are many ways of representing derivatives:

\[ y = mx+c \]

\[ \frac{d}{dx}[y] = m \]

Or you could also represent them like so:

\[ f(x) = mx + c \]

\[ f'(x) = m \]

Derivative at a point

Derivative at a point = gradient at a point = gradient of tangent

Positive gradient (line go up):

\[ \frac{dy}{dx} > 0 \]

Negative gradient (line go down):

\[ \frac{dy}{dx} < 0 \]

\[ \frac{dy}{dt} = \frac{dy}{dx} * \frac{dx}{dt} \]

Example:

\[ A = \textrm{Area of a circle} = \pi r^2 \]

Given constant rate of change of radius (\(\frac{dr}{dt} = 3\)), find rate of change of area when radius is at \(r\)

1. Formulate equation

\[ \frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt} \]

Rate of change of area \(=\) change in area with respect to radius \(\times\) rate of change of radius

2. Find derivatives

\[ \frac{dA}{dr} = 2\pi r \]

3. Solve for unknown

\[ \frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt} \]

\[ \frac{dA}{dt} = 2\pi r \times 3 \]

Substitute \(r\) for value specified in question

Nature of stationary points

Determine whether a stationary point (\(\frac{dy}{dx} = 0\)) is:
- Maximum point
- Minimum point
- Inflection point

There are two methods of finding this.

First derivative

Finding the gradient around the stationary point

Maximum point:

	\(f'(x)^-\)	\(f'(x)\)	\(f'(x)^+\)
Gradient	Positive gradient	0.00	Negative gradient
Direction of tangent line	/	-	\

Minimum point:

	\(f'(x)^-\)	\(f'(x)\)	\(f'(x)^+\)
Gradient	Negative gradient	0.00	Positive gradient
Direction of tangent line	\	-	/

Inflection point:

	\(f'(x)^-\)	\(f'(x)\)	\(f'(x)^+\)
Gradient	Negative gradient	0.00	Negative gradient
Direction of tangent line	\	-	\

	\(f'(x)^-\)	\(f'(x)\)	\(f'(x)^+\)
Gradient	Positive gradient	0.00	Positive gradient
Direction of tangent line	/	-	/

Second derivative

Simply find the derivative again:

\[ \frac{d}{dx}[\frac{dx}{dy}] = \frac{d^2y}{dx^2} \]

Or also written as:

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

After finding second derivative at stationary point:

If negative, then it is a maximum point
If positive, then it is a minimum point
If it is zero, then use first derivative test or unknown nature

Rules of differentiation

1. Power rule

Bring down power, decrease power

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

2. Chain rule

Bring down power, decrease power, differentiate inside

\[ \frac{d}{dx}[f(x)^n] = (n)(f(x)^{n-1})(f'(x)) \]

3. Product rule

\[ y = uv \]

\[ \frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx} \]

Differentiate u, leave v + Differentiate v, leave u

4. Quotient rule

\[ y = \frac{f(x)}{g(x)} \]

\[ \frac{dy}{dx} = \frac{f'(x)g(x) - g'(x)f(x)}{g(x)^2} \]

Trigonometric functions

Formula sheet will be given but still

\(y\)	\(\frac{dy}{dx}\)
\(sin(f(x))\)	\(f'(x)cos(f(x))\)
\(cos(f(x))\)	\(-f'(x)sin(f(x))\)
\(tan(f(x))\)	\(f'(x)sec^2(f(x))\)

Exponential functions

\(y\)	\(\frac{dy}{dx}\)
\(a^{f(x)}\)	\(a^{f(x)}(f'(x))(ln(a))\)
\(e^{f(x)}\)	\(f'(x)e^{f(x)}\)

Logarithmic functions

\(y\)	\(\frac{dy}{dx}\)
\(\log_{a}{f(x)}\)	\(\frac{f'(x)}{af(x)}\)
\(\ln{f(x)}\)	\(\frac{f'(x)}{f(x)}\)

Last update: June 11, 2023
Created: June 11, 2023