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 […]
August 19, 2024
Have you ever wondered how to take the derivative of an exponential function like \(2^x\) or \(4^{3x^2}\)? In this article, we will derive the formula for taking the derivative of any exponential function whatsoever. Derivation of the Formula In order to derive the formula for the derivative of an exponential function, we first need to […]
