October 7, 2024
The derivative of $e^{nx}$ is $\boxed{n\cdot e^{nx}}$. In this article, we will be exploring why. Recall that the Chain Rule states: \[\begin{align*} \dfrac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x), \end{align*}\] where $f(x)$ and $g(x)$ are functions. In this case, we let $f(x) = e^x$ and $g(x) = nx$ so that $f(g(x)) = e^{g(x)} = e^{nx}$. The next […]
October 7, 2024
We will use two important facts to write a proof for the derivative of $\cos x$. Firstly, we will use the angle sum formula $$\cos(x+y) = \cos x\cos y – \sin x\sin y$$ This formula is part of precalculus and is common knowledge. The second formula we will use is $$\lim_{h\to0}\frac{\sin h} h = 1$$ […]
October 7, 2024
At first glance, it looks like we should use the quotient rule to differentiate $\dfrac2x$. Indeed, we do see a fraction. Recall the quotient rule. For functions $f(x)$ and $g(x)$, $$\dfrac d{dx}\left(\frac {f(x)}{g(x)}\right) = \frac{f'(x)g(x) – f(x)g'(x)}{g^2(x)}$$ If we let $f(x) = 2$ and $g(x) = x$, then $f'(x) = 0$ and $g'(x) = 1$. […]
October 7, 2024
To find the derivative of 1/x, we rewrite it as $x^{-1}$. We may apply the power rule $\dfrac{d}{dx} x^n = nx^{n – 1}$: \[ \begin{align*} \dfrac{d}{dx} x^{-1} &= (-1) \cdot x^{-1-1}\\ \dfrac{d}{dx} x^{-1} &= -x^{-2} \end{align*} \] We get the end result that $\boxed{\dfrac{d}{dx} \dfrac1x = \frac{d}{dx} x^{-1} = -\dfrac1{x^2}}$. In fact, this technique generalizes […]
October 7, 2024
To find the derivative of $\dfrac{1}{1+x}$, we will apply the chain rule $$\dfrac{d}{dx}(f(g(x)) = f'(g(x))g'(x)$$ on the functions $f(x) = \dfrac1x$ and $g(x) = 1+x$. First, we need to find the derivatives of $f(x)$ and $g(x)$: $$f'(x) = \dfrac{d}{dx}(x^{-1}) = -x^{-2} = -\frac{1}{x^2}$$ and $$g'(x) = \dfrac{d}{dx}(x+1) = 1$$ We are now ready to apply […]
October 7, 2024
If you already know that the derivative of $\sin x$ is $\cos x$ and the derivative of $\cos x$ is $-\sin x$, then you can proceed to read this short article. If you wish to learn why $\dfrac d{dx}\sin x = \cos x$ and $\dfrac d{dx}\cos x = -\sin x$, you can refer to the […]
September 4, 2024
Understanding integrals and derivatives of trigonometric functions can be tough. For example, You may have been told that $\dfrac{d}{dx} \sin(x) = \cos(x)$, but not given a good explanation for it. In this article, we will show you with proof how to find the derivatives and integrals of $\cos x$ and $\sin x$ functions. Derivatives of […]
September 2, 2024
This article will teach you how to take the derivative of $4e^x$, as well as the proof for why the derivative is correct. Taking the Derivative Let us take the derivative of $\ln e^x$. Of course, $\ln e^x = x$, so the derivative should equal $1$. Using the chain rule, we get that $\frac{d}{dx} \ln […]
September 2, 2024
To find the derivative of 1, we will start by graphing the equation $y = 1$, pictured below: The derivative of a function at a point is the rate of change of a function at said specific point. In other words, it is the slope of the tangent line to the curve at a […]
