Practice Problems

Partial Derivatives

December 11, 2024

Introduction In calculus, we will often have to work with functions not in one variable, but in two or more. Functions of this type are called $\textbf{multivariable functions}$. Their notation is slightly different. Examples of multivariable functions are $f(x, y) = xy + 1$ and $f(x, y, z) = xy + yz$. The derivative of […]

Quotient Rule: Taking the Derivatives of Fractions

December 11, 2024

Introduction In calculus, derivatives are an extremely useful tool that are used in a variety of problems. In this article, we will specifically learn about taking the derivatives of fractions. Derivatives of Fractions A fraction typically appears as the ratio of two functions in the form $\dfrac{f(x)}{g(x)}$, where \( f(x) \) is the numerator and […]

Higher Order Differentiation

November 18, 2024

Introduction In calculus, taking the derivative of a function allows us to understand the rate of change of a function. This process allows us to gain information about the nature of a function, giving us a variety of applications to different problems. When we want to take the derivative of a function multiple times, we […]

Implicit Differentiation Guide

November 15, 2024

What is Implicit Differentiation? In many calculus problems, you’ll see equations that don’t exactly represent functions. Normally, we’re used to seeing equations like $$y = 2x + 1$$ where $y$ is isolated on one side. However, what if \(x\) and \(y\) are mixed together, like $$x^2 + y^2 = 1 { or } 3x^2y -4x\cos […]

Finding the Derivative of \(\ln x\)

November 7, 2024

Answer \[\begin{align*} \boxed{\frac{d}{dx}(\ln x) = \frac{1}{x}} \end{align*}\] The Derivative of \(\ln x\) The function $\ln x$ is called the natural logarithm function. More specifically, $\ln x = \ln_e x$, where $e \approx 2.718$. The derivative of $\ln x$ is: \[\begin{align*} \frac{d}{dx}(\ln x) = \frac{1}{x} \end{align*}\] However, some sources also use $\log x$ to describe $\ln […]

Derivative of Exponential Functions

November 7, 2024

Introduction In calculus, an exponential function refers to any function that is a base with an exponent of some expression of $x$. Examples include $f(x) = 2^x$, $3^x$, $e^{x – 1}$, $5^{\sqrt{x + 1}}$, and so on. Derivatives of exponential functions often appear in calculus. The Key Idea The derivative of an exponential function $f(x) […]

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