October 7, 2024
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 […]
October 7, 2024
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 […]
October 7, 2024
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 = […]
October 7, 2024
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} […]
October 7, 2024
Recall the power rule for taking derivatives of polynomials: $$\dfrac{d}{dx} x^n = nx^{n – 1}$$ The power rule works for all real values of $n$, not jus the integers. We may apply this rule to derive $\sqrt x = x^{\frac12}$. Using the rule above with $n =\frac12$ we get $$\dfrac{d}{dx} x^{\frac12} = \frac12 x^{-\frac12}$$ Usually, […]
October 7, 2024
The derivative of $\sin x$ is $\boxed{\cos x}$. We will use the limit definition of derivative to prove this. \[ \begin{align*} \frac{d}{dx} f(x) &= \lim_{h \to 0} \dfrac{f(x + h) – f(x)}h \end{align*} \] We will also use the trigonometric identity $\sin(a + b) = \cos a \sin b + \sin a \cos b$ (line […]
October 7, 2024
To find the derivative of $\ln x$, we will use implicit differentiation, which is a standard technique to find derivatives of inverse functions. Note that $\ln x$ is the inverse function of $e^x$, so it is natural for us to set $y = \ln x$. We know that $\dfrac{d}{dx} e^x = e^x$ and we will […]
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 $g'(x) = 1$. […]
