Derivatives - Page 4

Derivative of $3x$

To find the derivative of $3x$, we may apply the power rule which states $\dfrac{d}{dx} x^n = nx^{n – 1}$. In this case, \(3x\) can be rewritten as \(3x^1\), where the coefficient \(3\) is constant and \(x\) has an exponent of \(n=1\). Solution Using the power rule: \[\begin{align*} \dfrac{d}{dx} (3x) &= 3 \cdot \dfrac{d}{dx} (x^1)…

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Derivative of $\cot x$

The derivative of $\cot x$ is $\boxed{-\csc^2 x}$. We will use our knowledge of the derivatives of $\sin x$ and $\cos x$ to prove this result. Recall that \begin{align*} \dfrac d{dx} \sin x = \cos x\end{align*} and \begin{align*} \dfrac d{dx} \cos x = -\sin x\end{align*} For a detailed discussion and proof of the derivative of…

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Derivative of $\csc x$

The derivative of $\csc x$ is $\boxed{-\csc x\cot x}$. We will use our knowledge of the derivatives of $\sin x$ and $\cos x$ to prove this result. Recall that \begin{align*} \dfrac d{dx} \sin x = \cos x\end{align*} and \begin{align*} \dfrac d{dx} \cos x = -\sin x\end{align*} For a detailed discussion and proof of the derivative…

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Derivative of $\sec x$

The derivative of $\sec x$ is $\boxed{\sec x\tan x}$. We will use our knowledge of the derivatives of $\sin x$ and $\cos x$ to prove this result. Recall that \begin{align*} \dfrac d{dx} \sin x = \cos x\end{align*} and \begin{align*} \dfrac d{dx} \cos x = -\sin x\end{align*} For a detailed discussion and proof of the derivative…

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Derivative of $\tan x$

The derivative of $\tan x$ is $\boxed{\sec^2 x}$. We will use our knowledge of the derivatives of $\sin x$ and $\cos x$ to prove this result. Recall that \begin{align*} \dfrac d{dx} \sin x = \cos x\end{align*} and \begin{align*} \dfrac d{dx} \cos x = -\sin x\end{align*} For a detailed discussion and proof of the derivative of…

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Derivative of $\dfrac{x + 1}{x}$

The derivative of \( \dfrac{x+1}{x} \) is \( \boxed{-\dfrac{1}{x^2}} \). While we can use the \textbf{quotient rule}, splitting the expression into two terms provides a simpler approach. Method 1: Simplify First (Preferred Method) Rewrite \( \dfrac{x+1}{x} \) as: \[\begin{align*}\frac{x+1}{x} = \frac{x}{x} + \frac{1}{x} = 1 + \frac{1}{x}.\end{align*}\] Now differentiate term-by-term: \[\begin{align*}\frac{d}{dx} \left( 1 + \frac{1}{x}…

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Related Rates Ladder Problem
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Introduction One of the most common related rates problems is the ladder problem, which looks at how a ladder slides down a wall, assuming that the ladder always makes a right triangle with the wall. Let’s see how to solve these sorts of problems by working through a simple example. Example Let’s consider a ladder…

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Derivative of a Polar Equation

Polar equations can seem tricky, especially when it comes to taking derivatives. However, the derivative of a polar equation can be taken directly in polar coordinates without needing to convert to rectangular form. This article will guide you through the process with helpful examples. Which Derivative Are You Computing? In polar coordinates, you may want…

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Derivative Formula Sheet

Basic Derivative Formulas Constant Rule: $\dfrac{d}{dx}C=0$ Constant Multiple Rule: $\dfrac{d}{dx}\Big(Cf(x)\Big)=C\dfrac{d}{dx}f(x)$ Sum and Difference Rule: $\dfrac{d}{dx}\Big(f(x)\pm g(x)\Big)=\dfrac{d}{dx}f(x)\pm\dfrac{d}{dx}g(x)$ Power Rule: $\dfrac{d}{dx}x^n=nx^{n-1}$ Product Rule: $$\dfrac{d}{dx}\Big(f(x)g(x)\Big)=f(x)g'(x)+g(x)f'(x)$$ Quotient Rule: $$\dfrac{d}{dx}\left(\dfrac{f(x)}{g(x)}\right)=\dfrac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$$ Chain Rule: $$\dfrac{d}{dx}\Big(f(g(x))\Big)=f'(g(x))g'(x)$$ Where $f(x)$ is the outside function and $g(x)$ is the inside function. Exponential and Logarithmic Derivatives \[ \begin{array}{lll} \dfrac{d}{dx}e^x=e^x &&  \dfrac{d}{dx}a^x=a^x\ln{a} \\[10pt] \dfrac{d}{dx}\ln{x}=\dfrac{1}{x} && \dfrac{d}{dx}\log_a x=\dfrac{1}{x\ln{a}} \end{array}…

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Power Rule Proof

This article will walk you through how to prove the Power Rule and take the derivative of any polynomial. We are going to prove every step of the way. If you don’t want to see the proof and just want a formula, then just know that for any function $f(x) = x^n$, $f'(x) = nx^{n-1}$;…

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