Hao-Yu

Higher Order Differentiation

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…

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Implicit Differentiation Guide

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…

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Finding the Derivative of \(\ln x\)

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…

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Derivative of Exponential Functions

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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Derivative of exp

Introduction Derivative of “exp” means a derivative of $e^x$, where the number $e$ is an Euler’s constant, roughly equal to $2.71828$. As an equation, \[\begin{align*} \exp(x) = e^x, \text{where e} \approx 2.71828. \end{align*}\] Derivative of $e^x$ The derivative of the function is equal to the slope of the function at a given point. The derivative…

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