Introduction to Differential Calculus

Last updated Jul 29, 2023 Edit Source

# ← Methods Home

• Rates of change is the same as the gradient on a graph
  • It is the rate at which a value is increasing
• Gradient $(m)$ is $\frac{rise}{run}$
  • $m = \frac{y_2\\ -\\ y_1}{x_2\\ -\\ x_1}$
    • This equation calculates the rate of change between two points

• We know how to calculate the gradient (rate of change) between two points, but what happens when we want to find the gradient at one point, known as instantaneous rate of change
• Lets say we are looking to find the instantaneous change at point $a$ on the graph $f(x)$
  • Since we need two points to calculate gradient, we imagine point $b$ also on the graph of $f(x)$ as close to $a$ as possible
  • We can imagine a secant that goes through the points $a$ and $b$, this secant has a gradient that can be calculated traditionally, lets say the gradient is 1 500
    • As we move $b$ closer to $a$, the gradient will begin to approach the true value of the instantaneous rate of change
      • This value is known as the limiting value
    • Through this we can estimate the instantaneous rate of change, however this is still an approximation
      • Using the derivative we can calculate the true limiting value
  • For a function $y = f(x)$, the instantaneous change at the point $(a, f (a))$ is the gradient of the tangent line to the graph of $y = f(x)$ at the point $(a, f (a))$

• Textbook: Cambridge - 17C
• We looked at estimating the instantaneous rate of change at point $a$ by calculating the gradient between $a$ and $b$, while $b$ got closer and closer to $a$
  • Now we will examine the actual tangent at point $a$, the true instantaneous gradient
• The derivative is the gradient at any point for any graph
  • e.g. the derivative of $x^2$ ($f(x) = x^2$) is $2x$
  • See below for the full definition

Derivative Notation

The derivative of $f(x)$ is $f’(x)$

• The notation for the limit of $2x + h + 1$ as $h$ approached $0$ is ${\lim \atop h → 0}(2x\\ +\\ h\\ +\\ 1)$
• The derivative of a function with rule $f(x)$ may be found by:
  1. Finding an expression for the gradient of the line through $P(x, f(x))$ and $Q(x + h, f(x + h))$
  2. Finding the limit of this expression as $h$ approaches $0$
• Consider the function $f(x) = x^3$. By first finding the gradient of the secant through $P(2, 8)$ and $Q(2 + h, (2 + h)^3$, find the gradient of the tangent to the curve at the point $(2, 8)$.
  • Gradient of $PQ = \frac{(2\\ +\\ h)^2\\ -\\ 8}{2\\ +\\ h\\ -\\ 2}$
    • $= \frac{12h\\ +\\ 16h^2\\ +\\ h^3}{h}$
    • $= 12 + 6h + h^2$
  • The gradient of the tangent line at $(2, 8)$ is ${\lim \atop h → 0}(12\\ +\\ 6h\\ +\\ h^2) = 12$
    • Substitute $h = 0$ into the equation to find the gradient

• The derivative of function $f$ is denoted $f’$ and is defined by:
  • $f’(x) = {\lim \atop h → 0}\frac{f(x\\ +\\ h)\\ -\\ f(x)}{h}$
• The tangent line to a graph of the function $f$ at the point $(a, f(a))$ is defined to be the line through $(a, f(a))$ with gradient $f’(a)$

• There are certain rules that we observe in differentiation:
  • Constant function: If $f(x) = c$, then $f’(x) = 0$
  • Linear function: If $f (x) = mx + c$, then $f’(x) = m$
  • Multiple: If $f (x) = k \cdot g(x)$, where $k$ is a constant, then $f’(x) = k \cdot g’(x)$
    • The derivative of a number multiple is the multiple of the derivative
    • For example: if $f(x) = 5x^2$, then $f’(x) = 5(2x) = 10x$
  • Sum: If $f(x) = g(x) + h(x)$, then $f’(x) = g’(x) + h’(x)$ The derivative of the sum is the sum of the derivatives For example: if $f(x) = x^2 + 2x$, then $f’(x) = 2x + 2$
  • Difference: If $f(x) = g(x) - h(x)$, then $f’(x) = g’(x) - h’(x)$ The derivative of the difference is the difference of the derivatives For example: if $f(x) = x^2 - 2x$, then $f’(x) = 2x - 2$

The process of finding the derivative function is called differentiation

• For $f(x) = x^n → f’(x) = nx^{n-1}$ , where $n$ is a non-zero integer
  • This also works for negative powers
    • e.g. $f(x) = x^{-3} → f’(x) = -3x^{-4}$
• For $f(x) = c → f’(x) = 0$, where $c$ is a constant

Alternative Notation

• Lagrange Notation
  • The derivative of $f(x)$ is written as $f’(x)$
• Leibniz Notation
  • The derivative of $y = f(x)$ is written as $\frac{dy}{dx}$

• Textbook: Cambridge 17F

Estimating a Derivative Graph of a Polynomial Function

The graph of a derivative function for a polynomial will be the same as the polynomial graph of one less power

• e.g. the graph of $f’(x)$ when $f(x) = 2x^3 + 5x$ will be a parabola since the original function graph was a cubic

• A function $f$ is strictly increasing on an interval if $x_2 > x_1$ implies $f(x_2) > f(x_1)$
  • Strictly increasing means that the gradient cannot be negative at any point
    • $f’(x)$ can be $0$, as seen at the turning point in a cubic function
  • If $f’(x) > 0$ for all $x$ in the interval, then the function is strictly increasing
• A function $f$ is strictly decreasing on an interval if $x_2 > x_1$ implies $f(x_2) < f(x_1)$
  • Strictly decreasing means that the gradient cannot be positive at any point
    • $f’(x)$ can still be $0$ as explained above
  • If $f’(x) < 0$ for all $x$ in the interval, then the function is strictly decreasing

• Cambridge - 17G
• The process of finding a function from its derivative is known as antidifferentiation
  • e.g. the antiderivative for $f’(x) = 2x$ is $f(x) = x^2 + c$
    • The constant $c$ is what differs all possible functions for the derivative function $f’(x) = 2x$
  • There always are multiple antiderivatives for any function
• Integral Symbol - $\int$
  • Leibniz notation can be used to state the general antiderivative
    • e.g. $\int\\ 2x\\ dx = x^2 + c$
      • The general antiderivative OR indefinite integral of $2x$ with respect to $x$ is equal to $x^2 + c$
  • In general if $F’(x) = f(x)$, then $\int f(x)\\ dx = F(x) + c$, where $c$ is an arbitrary real number
    • General Formula: $\int x^n\\ dx = \frac{x^{n + 1}}{n\\ +\\ 1} + c$, $n \in \mathbb{N} \cup {0}$

• Sum: $\int f(x) + g(x)\\ dx = \int f(x)\\ dx + \int g(x)\\ dx$
• Difference: $\int f(x) - g(x)\\ dx = \int f(x)\\ dx - \int g(x)\\ dx$
• Multiple: $\int kf(x)\\ dx = k \int f(x)\\ dx$, where $k \in \mathbb{R}$
  • Do not multiply $c$ by $k$

Continue → Applications of Differential Calculus