∬Differential Calculus Unit 10 – Exponential and Trigonometric Derivatives

Key Concepts

• Exponential functions have the form $f(x) = a^x$, where $a$ is a positive constant not equal to 1
• Trigonometric functions include sine ($\sin x$), cosine ($\cos x$), tangent ($\tan x$), cosecant ($\csc x$), secant ($\sec x$), and cotangent ($\cot x$)
• The derivative of an exponential function $f(x) = a^x$ is $f'(x) = a^x \ln a$
• The derivatives of trigonometric functions are as follows:
  • $\frac{d}{dx} \sin x = \cos x$
  • $\frac{d}{dx} \cos x = -\sin x$
  • $\frac{d}{dx} \tan x = \sec^2 x$
  • $\frac{d}{dx} \csc x = -\csc x \cot x$
  • $\frac{d}{dx} \sec x = \sec x \tan x$
  • $\frac{d}{dx} \cot x = -\csc^2 x$
• The chain rule is used to find the derivative of a composite function, which is a function that is composed of two or more functions
• Implicit differentiation is a technique used to find the derivative of a function that is not explicitly defined as a function of $x$
• Real-world applications of exponential and trigonometric derivatives include population growth, radioactive decay, and simple harmonic motion

Exponential Functions and Their Derivatives

• Exponential functions have a constant base raised to a variable power, such as $f(x) = 2^x$ or $f(x) = e^x$
• The natural exponential function, denoted as $e^x$, has a base of $e \approx 2.71828$
• The derivative of an exponential function $f(x) = a^x$ is $f'(x) = a^x \ln a$, where $\ln$ is the natural logarithm
  • For the natural exponential function $e^x$, the derivative simplifies to $\frac{d}{dx} e^x = e^x$
• Exponential functions and their derivatives are used to model situations with constant percent growth or decay rates
  • Population growth: $P(t) = P_0 e^{kt}$, where $P_0$ is the initial population, $k$ is the growth rate, and $t$ is time
  • Radioactive decay: $N(t) = N_0 e^{-\lambda t}$, where $N_0$ is the initial amount, $\lambda$ is the decay constant, and $t$ is time
• The derivative of an exponential function is always proportional to the original function, with the proportionality constant being the natural logarithm of the base

Trigonometric Functions Review

• Trigonometric functions are based on the ratios of sides in a right triangle
• The sine function, $\sin x$, is the ratio of the opposite side to the hypotenuse
• The cosine function, $\cos x$, is the ratio of the adjacent side to the hypotenuse
• The tangent function, $\tan x$, is the ratio of the opposite side to the adjacent side
  • $\tan x = \frac{\sin x}{\cos x}$
• Cosecant ($\csc x$), secant ($\sec x$), and cotangent ($\cot x$) are reciprocal functions of sine, cosine, and tangent, respectively
  • $\csc x = \frac{1}{\sin x}$, $\sec x = \frac{1}{\cos x}$, and $\cot x = \frac{1}{\tan x}$
• Trigonometric functions are periodic, meaning they repeat their values at regular intervals
  • The sine and cosine functions have a period of $2\pi$, while the tangent function has a period of $\pi$
• Trigonometric identities, such as the Pythagorean identity ($\sin^2 x + \cos^2 x = 1$), are useful for simplifying expressions and solving equations

Derivatives of Trigonometric Functions

• The derivatives of the six basic trigonometric functions are as follows:
  • $\frac{d}{dx} \sin x = \cos x$
  • $\frac{d}{dx} \cos x = -\sin x$
  • $\frac{d}{dx} \tan x = \sec^2 x$
  • $\frac{d}{dx} \csc x = -\csc x \cot x$
  • $\frac{d}{dx} \sec x = \sec x \tan x$
  • $\frac{d}{dx} \cot x = -\csc^2 x$
• These derivatives can be derived using the definition of the derivative and trigonometric identities
• When differentiating trigonometric functions, it is essential to pay attention to the argument of the function (the expression inside the parentheses)
  • For example, $\frac{d}{dx} \sin(2x) = 2\cos(2x)$ by the chain rule
• Derivatives of trigonometric functions are used in various applications, such as finding the slope of a curve at a given point or analyzing the motion of objects in simple harmonic motion

Chain Rule Applications

• The chain rule is used to find the derivative of a composite function, which is a function that is composed of two or more functions
• If $f(x)$ and $g(x)$ are differentiable functions, then the derivative of their composite function $f(g(x))$ is given by:
  • $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$
• The chain rule can be applied multiple times for functions with more than two nested functions
• When applying the chain rule to exponential and trigonometric functions, it is essential to identify the inner and outer functions correctly
  • Example: $\frac{d}{dx} e^{\sin x} = e^{\sin x} \cdot \cos x$
  • Example: $\frac{d}{dx} \tan(e^x) = \sec^2(e^x) \cdot e^x$
• The chain rule is also used in implicit differentiation and in finding the derivatives of inverse functions

Implicit Differentiation

• Implicit differentiation is a technique used to find the derivative of a function that is not explicitly defined as a function of $x$
• In implicit differentiation, we differentiate both sides of an equation with respect to $x$, treating $y$ as a function of $x$
• The process involves applying the chain rule to terms containing $y$, since $y$ is a function of $x$
  • Example: Given $x^2 + y^2 = 1$, find $\frac{dy}{dx}$
    1. Differentiate both sides with respect to $x$: $\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(1)$
    2. Apply the chain rule to the term containing $y$: $2x + 2y \frac{dy}{dx} = 0$
    3. Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = -\frac{x}{y}$
• Implicit differentiation is useful for finding the derivatives of functions defined by equations that are not easily solved for $y$ in terms of $x$

Real-World Applications

• Exponential and trigonometric derivatives have numerous real-world applications in various fields, such as physics, biology, and economics
• Population growth and decay: Exponential functions model the growth or decline of populations over time
  • Example: Bacterial growth in a petri dish can be modeled by $P(t) = P_0 e^{kt}$, where $P_0$ is the initial population, $k$ is the growth rate, and $t$ is time
• Radioactive decay: The decay of radioactive substances follows an exponential decay model
  • Example: The amount of a radioactive isotope remaining after time $t$ is given by $N(t) = N_0 e^{-\lambda t}$, where $N_0$ is the initial amount, $\lambda$ is the decay constant, and $t$ is time
• Simple harmonic motion: Trigonometric functions describe the periodic motion of objects, such as springs and pendulums
  • Example: The displacement of a mass on a spring can be modeled by $x(t) = A \cos(\omega t + \phi)$, where $A$ is the amplitude, $\omega$ is the angular frequency, $t$ is time, and $\phi$ is the phase shift
• Alternating current: Trigonometric functions are used to model the voltage and current in alternating current (AC) circuits
  • Example: The voltage in an AC circuit can be described by $V(t) = V_0 \sin(\omega t)$, where $V_0$ is the peak voltage, $\omega$ is the angular frequency, and $t$ is time

Common Mistakes and How to Avoid Them

• Forgetting to apply the chain rule when differentiating composite functions
  • Always identify the inner and outer functions and apply the chain rule accordingly
• Incorrectly differentiating trigonometric functions
  • Memorize the derivatives of the six basic trigonometric functions and pay attention to the argument of the function
• Misapplying the chain rule in implicit differentiation
  • Remember to treat $y$ as a function of $x$ and apply the chain rule to terms containing $y$
• Confusing the properties of exponential and trigonometric functions
  • Exponential functions have a constant base raised to a variable power, while trigonometric functions are based on the ratios of sides in a right triangle
• Neglecting to simplify the final answer
  • Simplify your answer by combining like terms, factoring, or using trigonometric identities when appropriate
• Not checking the domain of the function before differentiating
  • Ensure that the function is differentiable on its domain and consider any restrictions on the variable