Derivatives - Page 5

Review of Derivatives

Hello everyone! In this blog, we will be reviewing everything about derivatives that you need to know for the AP Calculus test, from the definition of a derivative to more advanced topics like implicit differentiation. This blog is for the curious who want to see the proofs of the differentiation rules we all use. Without…

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Derivative of Square Root of x

At first, it may appear daunting to calculate the derivative of $\sqrt x$. Indeed, you have not seen anything like this before! You know the power rule, the derivatives of trigonometric functions, like derivative of sine and derivative of cosine. You might even remember the derivatives of tangent, cotangent, secant and cosecant. Unexpectedly, even the…

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1/x-1 Derivative

To find the derivative of $\dfrac{1}{x-1}$, we have to use the power rule and the chain rule. Step 1. Rewrite First, it is easier to rewrite $\dfrac{1}{x-1}$ as $(x-1)^{-1}$. This makes differentiation simpler because we can now apply the power rule directly. Step 2. Power Rule According to the power rule, $\dfrac{d}{dx} x^n = n…

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Derivative of a negative $x$

\[\begin{align*} \boxed{\displaystyle \frac{d}{dx} (-x) = -1} \end{align*}\] Let’s learn how to find the derivative of the function \( f(x) = -x \). Don’t worry—we’ll keep it simple and easy to understand! What is a Derivative? A derivative tells us how a function changes as the input \( x \) changes. Think of it like measuring how…

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Derivative of f(x)=x

\[\begin{align*} \boxed{\displaystyle \frac{d}{dx} (x) = 1} \end{align*}\] Let’s learn how to find the derivative of the function $f(x) = x$. It’s simple, we will show every step. What is a Derivative? A derivative tells us how a function changes as the input $x$ changes. Think of it like finding out how fast or slow something is…

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Derivative of 4 ^ x or $4^x$

Introduction Hello everyone! In this article, we will review how to take the derivative of $4^x$. This is a very instructive example for how to take the derivative of an exponential function with a base other than $e$, which is a very important concept to understand. Without further ado, let’s get into it! Taking the…

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Limit Definition of Derivative

The limit definition of a derivative for a function $f(x)$is as follows: \[ \begin{align*} \lim_{h\to0} \dfrac{f(x+h) – f(x)}{h} \end{align*} \] To understand this function, imagine two points. Both are on the $f(x)$ curve, and one is at $(x,f(x))$, while the other is at $(x+h,f(x+h)$. What this function says as the second point approaches the first…

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Derivative of x ln x

The derivative of x lnx is $\boxed{\frac{d}{dx} (x\ln x) = \ln x + 1}$. To show this, we will use the product rule, which states that for two functions $f(x)$ and $g(x)$ \begin{align*} &\dfrac{d}{dx} \bigg(f(x)g(x)\bigg) = f'(x)g(x) + f(x)g'(x) \end{align*} In our case, we will let \(f(x)=x\) and \(g(x)=\ln x\). Then the product rule states…

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Derivative of $\dfrac{x-2}{x-1}$

To find the derivative of $\dfrac{x-2}{x-1}$, we use the quotient rule, which states that for two functions $f(x)$ and $g(x)$, provided that $g(x)$ is not equal to $0$ and that both derivatives of $f(x)$ and $g(x)$ exist, $$\dfrac{d}{dx} \dfrac{f(x)}{g(x)} = \dfrac{f'(x)g(x) – g'(x)f(x)}{g(x)^2}$$ We will use the fact that $$\dfrac{d}{dx} (x+1) = \dfrac{d}{dx} x+\dfrac{d}{dx}1 =…

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Derivative of Trig Functions

The derivatives of trig functions are a core part of solving many Calculus problems! They are listed below: \[ \begin{align*} \dfrac{d}{dx} \sin x &= \cos x\\ \dfrac{d}{dx} \cos x &= -\sin x\\ \dfrac{d}{dx} \tan x &= \sec^2 x\\ \dfrac{d}{dx} \cot x &= – \csc^2 x \\ \dfrac{d}{dx} \sec x &= \sec x \tan x\\ \dfrac{d}{dx}…

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