Derivatives - Page 2

Derivative of $\dfrac3x$

\[\begin{align*} \boxed{\dfrac{d}{dx} \left(\dfrac3x\right) = -\dfrac3{x^2}} \end{align*}\] Solving for the Derivative We may pull the constant, $3$, out of the derivative and focus on $\dfrac{d}{dx} \left(\dfrac1x\right)$. This is the same as $x^{-1}$, so we may apply the power rule: \[\begin{align*} \dfrac{d}{dx} x^n = nx^{n-1} \end{align*}\] Doing this with $x^{-1}$, we get our derivative is equal to…

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Derivative of $\sec(2x)$

\[\begin{align*} \boxed{\dfrac{d}{dx} \sec(2x) = 2 \cdot \sec \left(2x\right)\tan \left(2x\right)} \end{align*}\] Solving for the Derivative Applying the chain rule: \[\begin{align*} \dfrac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)) \end{align*}\] We may set our $g(x) = 2x$ and our $f(x) = \sec x$. Doing this, we get that \[\begin{align*} \dfrac{d}{dx} \sec(2x) &= \sec \left(2x\right)\tan \left(2x\right)\frac{d}{dx}\left(2x\right)\\ &= \boxed{2 \cdot \sec \left(2x\right)\tan…

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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 2x ln(x)

\[\begin{align*} \boxed{\frac{d}{dx} \left( 2x \ln(x) \right) = 2 \ln(x) + 2} \end{align*}\] Introduction To find the derivative of \( 2x \ln(x) \), we will use the product rule. Learning to work with logarithmic expressions is important in calculus as they show up later when tackling integration. Step-by-Step Solution The product rule, which states that if…

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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 $e^x$

In this article, we are going to prove that the derivative of $e^x$ is equal to $\boxed{e^x}$. To find the derivative of $e^x$, we must first define $e$. The formal definition of $e$ is: \[\begin{align*} e = \lim_{n\to\infty}(1+\dfrac1n)^n \end{align*}\] $e$ is approximately equal to $2.71828$. This value does not need to me memorized, as it…

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Understanding the Derivative of \( 2^x \)

\[\begin{align*} \boxed{\frac{d}{dx} \left( 2^x \right) = 2^x \ln(2)} \end{align*}\] Introduction: Finding the derivative of \( 2^x \) helps us understand how the function changes as \( x \) changes. Unlike derivatives of polynomials or basic trigonometric functions, differentiating an exponential function involves the natural logarithm. Let’s walk through the steps. Step-by-Step Solution: To differentiate \(…

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Derivative of a^x

The derivative of $a^x$, where $a$ is a constant is $\dfrac{d}{dx}(a^x) = \boxed{a^x \cdot \ln a}$. Let’s examine why this is true. Finding the Derivative of $a^x$: We will start by rewriting $a^x$ as: \[\begin{align*} a^x = (e^{\ln a})^x = e^{\ln a \cdot x}. \end{align*}\] Now, recall the formula $\dfrac{d}{dx}(e^{nx}) = e^{nx} \cdot n$. In…

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