Calculus Classrooms
52.1 Calculus Introduction
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It has two major branches: differential calculus and integral calculus.
- Differential Calculus: Deals with rates of change and slopes of curves. It's about finding how quickly quantities change.
- Integral Calculus: Deals with accumulation of quantities and the areas under and between curves. It's about combining small pieces to find the total.
Calculus is fundamental to understanding motion, growth, decay, optimization, and many other real-world phenomena in science, engineering, economics, and medicine.
Which branch of calculus is concerned with rates of change?
52.2 Functional Notation
In calculus, functions are usually written using functional notation. For example, $f(x)$ denotes a function $f$ that takes an input $x$. If $y$ is a function of $x$, we can write: $y = f(x)$.
Derivatives can be expressed in several notations:
- Leibniz Notation: $\frac{dy}{dx}$ — pronounced “dee-y over dee-x”, showing the derivative of $y$ with respect to $x$.
- Lagrange (Prime) Notation: $f'(x)$ — read as “f prime of x”, representing the derivative of $f$.
- Newton (Dot) Notation: $\dot{y}$ — typically used in physics for time derivatives.
Understanding these notations is essential for interpreting and working with derivatives.
52.3 Gradient of a Curve
The gradient (or slope) of a straight line is constant. However, for a curve, the gradient changes from point to point. The gradient of a curve at a specific point is defined as the gradient of the tangent line to the curve at that point.
Differential calculus provides a method to find this instantaneous rate of change. The derivative of a function $y = f(x)$, denoted as $\frac{dy}{dx}$ or $f'(x)$, gives a formula for the gradient of the curve at any point $x$.
For instance, if $y = x^2$, then $\frac{dy}{dx} = 2x$. At $x=1$, the gradient is $2(1)=2$. At $x=3$, the gradient is $2(3)=6$.
52.4 Differentiation from First Principles
Differentiation from first principles, also known as the delta method or limit definition of the derivative, defines the derivative of a function $f(x)$ as:
$$ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$This formula represents the gradient of the secant line between $x$ and $x+h$ as $h$ approaches zero, effectively becoming the gradient of the tangent line at $x$. While other differentiation rules are more efficient for complex functions, understanding first principles is crucial for grasping the fundamental concept of the derivative.
52.5 Rule for $y = ax^n$
The power rule is one of the most fundamental rules of differentiation. For a function of the form $y = ax^n$, where $a$ is a constant coefficient and $n$ is a real number exponent, its derivative with respect to $x$ is given by:
$$ \frac{dy}{dx} = n \cdot ax^{n-1} $$This rule states that you multiply the coefficient $a$ by the exponent $n$, and then decrease the exponent by $1$. For example, if $y = 3x^4$, then $\frac{dy}{dx} = 4 \cdot 3x^{4-1} = 12x^3$. If $y = x$, then $y = 1x^1$, so $\frac{dy}{dx} = 1 \cdot 1x^{1-1} = x^0 = 1$. If $y = C$ (a constant), it can be seen as $y = Cx^0$, so $\frac{dy}{dx} = 0 \cdot Cx^{-1} = 0$.
52.6 Differentiating $\sin$ & $\cos$
The derivatives of trigonometric functions are essential in calculus, especially when dealing with oscillatory phenomena. The two most fundamental are:
- The derivative of $\sin(x)$ with respect to $x$ is $\cos(x)$: $$ \frac{d}{dx}(\sin x) = \cos x $$
- The derivative of $\cos(x)$ with respect to $x$ is $-\sin(x)$: $$ \frac{d}{dx}(\cos x) = -\sin x $$
These derivatives can be derived from first principles using trigonometric identities and limits. Remember these basic forms as they are foundational for more complex trigonometric differentiation.
52.7 Differentiating $e^{ax}$ & $\ln ax$
Exponential and natural logarithmic functions are also crucial in calculus. Their differentiation rules are:
- The derivative of $e^{ax}$ with respect to $x$ is $ae^{ax}$: $$ \frac{d}{dx}(e^{ax}) = ae^{ax} $$ This shows that $e^x$ is its own derivative ($a=1$).
- The derivative of $\ln(ax)$ with respect to $x$ is $\frac{1}{x}$: $$ \frac{d}{dx}(\ln(ax)) = \frac{1}{x} $$ Note that the $a$ cancels out, as $\ln(ax) = \ln a + \ln x$, and $\ln a$ is a constant.
These rules are widely applied in problems involving natural growth and decay, continuous compounding, and various scientific models.
Methods of Differentiation
53.1 Common Functions
When differentiating sums or differences of functions, the derivative of the sum/difference is simply the sum/difference of their individual derivatives. This is known as the linearity property of differentiation.
$$ \frac{d}{dx}[f(x) \pm g(x)] = \frac{d}{dx}[f(x)] \pm \frac{d}{dx}[g(x)] $$This allows us to combine the power rule, trigonometric derivatives, and exponential/logarithmic derivatives when functions are added or subtracted.
53.2 Product Rule
The product rule is used to differentiate functions that are the product of two other functions. If $y = u(x)v(x)$, where $u$ and $v$ are differentiable functions of $x$, then the product rule states:
$$ \frac{dy}{dx} = u'v + uv' $$where $u'$ is the derivative of $u$ with respect to $x$, and $v'$ is the derivative of $v$ with respect to $x$. This rule is essential when you cannot easily expand the product into simpler terms.
53.3 Quotient Rule
The quotient rule is used to differentiate functions that are the ratio of two other functions. If $y = \frac{u(x)}{v(x)}$, where $u$ and $v$ are differentiable functions of $x$ and $v(x) \neq 0$, then the quotient rule states:
$$ \frac{dy}{dx} = \frac{u'v - uv'}{v^2} $$This rule is often remembered as "low dee high minus high dee low, over low squared" (where 'low' is the denominator $v$, and 'high' is the numerator $u$). It's indispensable for differentiating rational functions.
53.4 Chain Rule (Function of a Function)
The chain rule is used to differentiate composite functions. If $y$ is a function of $u$, and $u$ is a function of $x$ (i.e., $y = f(u(x))$), then the chain rule states:
$$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} $$This rule is used when you have a function "inside" another function, such as $(2x+1)^3$ or $\sin(x^2)$. It's one of the most powerful and frequently used rules in differentiation.
53.5 Successive Differentiation
Successive differentiation involves finding the derivative of a derivative. The first derivative, $\frac{dy}{dx}$ or $f'(x)$, gives the rate of change of the original function. The second derivative, $\frac{d^2y}{dx^2}$ or $f''(x)$, gives the rate of change of the first derivative (e.g., acceleration if the first derivative is velocity).
You can continue this process to find third, fourth, and higher-order derivatives. For example, the third derivative is $\frac{d^3y}{dx^3}$ or $f'''(x)$.