November 1, 2024
\[\begin{align*} \boxed{\frac{d}{dx} \left( 2^x \right) = 2^x \ln(2)} \end{align*}\] 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 \( […]
October 25, 2024
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 […]
October 25, 2024
The derivative of $\dfrac{x}{5}$ is $\boxed{\dfrac{1}{5}}$. To find the derivative of $\dfrac{x}{5}$, we may apply the power rule: \[\begin{align*} \dfrac{d}{dx} x^n = nx^{n – 1} \end{align*}\] In this case, $\dfrac{x}{5}$ can be rewritten as $\dfrac{1}{5} \cdot x^1$, where the coefficient $\dfrac15$ is constant and \(x\) has an exponent of \(n=1\). Solution Using the power rule: […]
October 25, 2024
The derivative of $\dfrac{x}{3}$ is $\boxed{\dfrac{1}{3}}$. To find the derivative of $\dfrac{x}{3}$, we may apply the power rule: \[\begin{align*} \dfrac{d}{dx} x^n = nx^{n – 1} \end{align*}\] In this case, $\dfrac{x}{3}$ can be rewritten as $\dfrac{1}{3} \cdot x^1$, where the coefficient $\dfrac13$ is constant and \(x\) has an exponent of \(n=1\). Solution Using the power rule: […]
October 25, 2024
The derivative of $5x$ is $\boxed{5}$. To find the derivative of $5x$, we may apply the power rule: \[\begin{align*} \dfrac{d}{dx} x^n = nx^{n – 1} \end{align*}\] In this case, \(5x\) can be rewritten as \(5x^1\), where the coefficient \(5\) is constant and \(x\) has an exponent of \(n=1\). Solution Using the power rule: \[\begin{align*} \dfrac{d}{dx} […]
October 25, 2024
The derivative of $4x$ is $\boxed{4}$. To find the derivative of $4x$, we may apply the power rule: \[\begin{align*} \dfrac{d}{dx} x^n = nx^{n – 1} \end{align*}\] In this case, \(4x\) can be rewritten as \(4x^1\), where the coefficient \(4\) is constant and \(x\) has an exponent of \(n=1\). Solution Using the power rule: \[\begin{align*} \dfrac{d}{dx} […]
October 25, 2024
The derivative of $\dfrac1{1 + \sin x}$ is $\boxed{-\dfrac{\cos x}{(1 + \sin x)^2}}$. To find the derivative of $\dfrac1{1 + \sin x}$, we will use a combination of the power rule and chain rule. The power rule is: \[\begin{align*} \dfrac{d}{dx} x^n &= nx^{n – 1} \end{align*}\] The chain rule is: \[\begin{align*} \dfrac{d}{dx}(f(g(x))) = f'(g(x)) \cdot […]
October 25, 2024
The derivative of $\dfrac1{x^2}$ is $\boxed{-\dfrac2{x^3}}$. To find the derivative of $\dfrac1{x^2}$, we will use the Power Rule: \[\begin{align*} \dfrac{d}{dx} x^n &= nx^{n – 1} \end{align*}\] Notice that $\dfrac1{x^2}$ may be rewritten as $x^{-2}$, making the use of the power rule quite straightforward. \[\begin{align*} \dfrac{d}{dx} \dfrac1{x^2} &= \dfrac{d}{dx} x^{-2}\\ &= -2x^{-2 – 1}\\ &= -2x^{-3}\\ […]
October 22, 2024
There are a couple of notations when it comes to denoting the derivative of functions. The most common notations you will see include: $\bullet$ Leibniz Notation: $\dfrac{d}{dx} f(x)$ $\bullet$ Lagrange Notation: $f'(x)$ $\bullet$ Euler Notation: $D f(x)$ $\bullet$ Newton Notation: $\dot y$ Most likely, you will find Leibniz and Lagrange’s notations being used interchangeably in […]
