Calculus Study Notes: Differentiation, Tangent Lines, and Implicit Differentiation

Differentiation of Functions

Basic Differentiation Rules

Differentiation is a fundamental concept in calculus, used to find the rate at which a function changes. The derivative of a function describes its instantaneous rate of change with respect to its variable.

• Power Rule: For $y = x^n$, the derivative is $\frac{dy}{dx} = n x^{n-1}$.

• Exponential Functions: For $y = e^x$, the derivative is $\frac{dy}{dx} = e^x$.

• Logarithmic Functions: For $y = \ln(x)$, the derivative is $\frac{dy}{dx} = \frac{1}{x}$.

• Product Rule: For $y = u(x) v(x)$, $\frac{dy}{dx} = u'(x) v(x) + u(x) v'(x)$.

• Quotient Rule: For $y = \frac{u(x)}{v(x)}$, $\frac{dy}{dx} = \frac{u'(x) v(x) - u(x) v'(x)}{[v(x)]^2}$.

Example: For $y = x^2 e^x$, use the product rule:

$\frac{dy}{dx} = 2x e^x + x^2 e^x$

Derivative of Composite Functions

When differentiating composite functions, the chain rule is used. If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$.

• Example: $y = \sin(x^2)$ $\frac{dy}{dx} = \cos(x^2) \cdot 2x$

Finding Tangent Lines

Equation of the Tangent Line

The tangent line to a curve at a point $x=a$ is given by:

$y = f(a) + f'(a)(x - a)$

• Example: For $y = 2 \arcsin\left(\frac{\pi x}{4}\right)$ at $x=1$:

• Find $y(1)$ and $y'(1)$:

• $y(1) = 2 \arcsin\left(\frac{\pi}{4}\right)$

• $y'(x) = 2 \cdot \frac{1}{\sqrt{1 - \left(\frac{\pi x}{4}\right)^2}} \cdot \frac{\pi}{4}$

• At $x=1$, $y'(1) = \frac{2\pi}{4\sqrt{1 - (\frac{\pi}{4})^2}} = \frac{\pi}{\sqrt{16 - \pi^2}}$

• Equation: $y = 2 \arcsin\left(\frac{\pi}{4}\right) + \frac{\pi}{\sqrt{16 - \pi^2}} (x - 1)$

Additional info: The tangent line formula is derived from the point-slope form using the value and derivative at the point of tangency.

Implicit Differentiation

Definition and Application

Implicit differentiation is used when a function is not given explicitly as $y = f(x)$, but rather as a relationship between $x$ and $y$.

• Example: For $\theta^{1/2} + r^{1/2} = 1$, differentiate both sides with respect to $\theta$:

• $\frac{1}{2} \theta^{-1/2} \frac{d\theta}{d\theta} + \frac{1}{2} r^{-1/2} \frac{dr}{d\theta} = 0$

• Simplify: $\frac{1}{2} \theta^{-1/2} + \frac{1}{2} r^{-1/2} \frac{dr}{d\theta} = 0$

• Solving for $\frac{dr}{d\theta}$: $\frac{dr}{d\theta} = -\frac{\theta^{-1/2}}{r^{-1/2}}$

Additional info: Implicit differentiation is essential for finding derivatives of curves defined by equations involving both $x$ and $y$.

Special Derivatives and Applications

Logarithmic and Exponential Functions

• Example: $y = \ln\left(\frac{x}{2}\right)$ $\frac{dy}{dx} = \frac{1}{x}$

• Example: $y = e^{x \ln(\sin x)}$ $\frac{dy}{dx} = e^{x \ln(\sin x)} \left[ \ln(\sin x) + x \frac{\cos x}{\sin x} \right]$

Functions with Variable Exponents

• Example: $y = (\sin x)^x$ Use logarithmic differentiation: $\ln y = x \ln(\sin x)$ $\frac{1}{y} \frac{dy}{dx} = \ln(\sin x) + x \frac{\cos x}{\sin x}$ $\frac{dy}{dx} = (\sin x)^x \left[ \ln(\sin x) + x \frac{\cos x}{\sin x} \right]$

Summary Table: Differentiation Rules

Function	Derivative	Rule Used
$x^n$	$n x^{n-1}$	Power Rule
$e^x$	$e^x$	Exponential Rule
$\ln(x)$	$\frac{1}{x}$	Logarithmic Rule
$u(x) v(x)$	$u'(x) v(x) + u(x) v'(x)$	Product Rule
$\frac{u(x)}{v(x)}$	$\frac{u'(x) v(x) - u(x) v'(x)}{[v(x)]^2}$	Quotient Rule
$f(g(x))$	$f'(g(x)) \cdot g'(x)$	Chain Rule
Implicit: $F(x, y) = 0$	$\frac{dy}{dx}$ via implicit differentiation	Implicit Rule