From fb87077d38c5c63c29573023e376fd7ee00b1712 Mon Sep 17 00:00:00 2001 From: saeld13 Date: Fri, 2 May 2025 21:35:00 -0500 Subject: [PATCH 1/5] change --- calculus/derivatives.md | 13 +++++++++++++ 1 file changed, 13 insertions(+) diff --git a/calculus/derivatives.md b/calculus/derivatives.md index f8c7ba028..d40fdc317 100644 --- a/calculus/derivatives.md +++ b/calculus/derivatives.md @@ -1 +1,14 @@ # Derivatives +$$ +\begin{array}{lll} +\text{Rule} & f(x) & f'(x) \\ +\hline +\text{Constant} & k & 0 \\ +\text{Power} & x^n & nx^{n-1} \\ +\text{Multiple} & k u(x) & k u'(x) \\ +\text{Sum} & u(x) + v(x) & u'(x) + v'(x) \\ +\text{Difference} & u(x) - v(x) & u'(x) - v'(x) \\ +\text{Exponential} & a^x & \frac{a^x}{\ln a} \\ +\text{Logarithm} & \log_a x & \frac{1}{x \ln a} \\ +\end{array} +$$ \ No newline at end of file From 6e9f33c9c1ff03c6de81de92e7149095cd7a4d2c Mon Sep 17 00:00:00 2001 From: saeld13 Date: Tue, 6 May 2025 10:19:23 -0500 Subject: [PATCH 2/5] change --- calculus/derivatives.md | 22 +++++++++++++++++++++- 1 file changed, 21 insertions(+), 1 deletion(-) diff --git a/calculus/derivatives.md b/calculus/derivatives.md index d40fdc317..ac5fc1250 100644 --- a/calculus/derivatives.md +++ b/calculus/derivatives.md @@ -11,4 +11,24 @@ $$ \text{Exponential} & a^x & \frac{a^x}{\ln a} \\ \text{Logarithm} & \log_a x & \frac{1}{x \ln a} \\ \end{array} -$$ \ No newline at end of file +$$ + +text{(1) } f(x) = x^3 \quad \rightarrow \quad f'(x) = 3x^{3-1} = 3x^2 + +\text{(2) } f(x) = -x^2 + 3 \quad \rightarrow \quad f'(x) = \frac{d}{dx}(-x^2) + \frac{d}{dx}(3) \\ += -\frac{d}{dx}(x^2) + \frac{d}{dx}(3) \\ += -2x^{2-1} + 0 \\ += -2x + +\text{(3) } f(x) = \frac{4x^3}{3} - x + 2e^x \quad \rightarrow \quad f'(x) = \frac{4}{3}\frac{d}{dx}(x^3) - \frac{d}{dx}(x) + 2\frac{d}{dx}(e^x) \\ += \frac{4}{3}(3x^2) -1 + 2e^x \\ += 4x^2 -1 +2e^x + +\text{(4) } f(x) = \frac{3}{x^2} + \sqrt{x} \quad \rightarrow \quad f'(x) = 3\frac{d}{dx}(x^{-2}) + \frac{d}{dx}(x^{1/2}) \\ += 3(-2x^{-3}) + \frac{1}{2}x^{-1/2} \\ += \frac{-6}{x^3} + \frac{1}{2\sqrt{x}} + +\text{(5) } f(x) = \log(4x) \quad \rightarrow \quad f'(x) = \frac{d}{dx}[\log 4 + \log x] \\ += \frac{d}{dx}(\log 4) + \frac{d}{dx}(\log x) \\ += 0 + \frac{1}{x \ln 10} \\ += \frac{1}{x \ln 10} \ No newline at end of file From 4d9602810842af927259773bbb452828ae36b2c9 Mon Sep 17 00:00:00 2001 From: saeld13 Date: Wed, 7 May 2025 12:09:59 -0500 Subject: [PATCH 3/5] change --- calculus/derivatives.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/calculus/derivatives.md b/calculus/derivatives.md index ac5fc1250..a846b76ab 100644 --- a/calculus/derivatives.md +++ b/calculus/derivatives.md @@ -13,6 +13,7 @@ $$ \end{array} $$ +$$ text{(1) } f(x) = x^3 \quad \rightarrow \quad f'(x) = 3x^{3-1} = 3x^2 \text{(2) } f(x) = -x^2 + 3 \quad \rightarrow \quad f'(x) = \frac{d}{dx}(-x^2) + \frac{d}{dx}(3) \\ @@ -31,4 +32,5 @@ text{(1) } f(x) = x^3 \quad \rightarrow \quad f'(x) = 3x^{3-1} = 3x^2 \text{(5) } f(x) = \log(4x) \quad \rightarrow \quad f'(x) = \frac{d}{dx}[\log 4 + \log x] \\ = \frac{d}{dx}(\log 4) + \frac{d}{dx}(\log x) \\ = 0 + \frac{1}{x \ln 10} \\ -= \frac{1}{x \ln 10} \ No newline at end of file += \frac{1}{x \ln 10} +$$ \ No newline at end of file From 622918ce5dacf1c2f798e3a28369791399196e67 Mon Sep 17 00:00:00 2001 From: saeld13 Date: Wed, 7 May 2025 16:30:23 -0500 Subject: [PATCH 4/5] change --- calculus/derivative-rules.md | 35 ++++++++++++++++------------------- 1 file changed, 16 insertions(+), 19 deletions(-) diff --git a/calculus/derivative-rules.md b/calculus/derivative-rules.md index 9d1712e99..7eb3bad73 100644 --- a/calculus/derivative-rules.md +++ b/calculus/derivative-rules.md @@ -15,25 +15,22 @@ $$ $$ $$ -text{(1) } f(x) = x^3 \quad \rightarrow \quad f'(x) = 3x^{3-1} = 3x^2 - -\text{(2) } f(x) = -x^2 + 3 \quad \rightarrow \quad f'(x) = \frac{d}{dx}(-x^2) + \frac{d}{dx}(3) \\ -= -\frac{d}{dx}(x^2) + \frac{d}{dx}(3) \\ -= -2x^{2-1} + 0 \\ -= -2x - -\text{(3) } f(x) = \frac{4x^3}{3} - x + 2e^x \quad \rightarrow \quad f'(x) = \frac{4}{3}\frac{d}{dx}(x^3) - \frac{d}{dx}(x) + 2\frac{d}{dx}(e^x) \\ -= \frac{4}{3}(3x^2) -1 + 2e^x \\ -= 4x^2 -1 +2e^x - -\text{(4) } f(x) = \frac{3}{x^2} + \sqrt{x} \quad \rightarrow \quad f'(x) = 3\frac{d}{dx}(x^{-2}) + \frac{d}{dx}(x^{1/2}) \\ -= 3(-2x^{-3}) + \frac{1}{2}x^{-1/2} \\ -= \frac{-6}{x^3} + \frac{1}{2\sqrt{x}} - -\text{(5) } f(x) = \log(4x) \quad \rightarrow \quad f'(x) = \frac{d}{dx}[\log 4 + \log x] \\ -= \frac{d}{dx}(\log 4) + \frac{d}{dx}(\log x) \\ -= 0 + \frac{1}{x \ln 10} \\ -= \frac{1}{x \ln 10} +\begin{align*} +\text{(1)} \quad f(x) &= x^3 \quad \to \quad f'(x) = 3x^{3-1} = 3x^2 \\[10pt] +\text{(2)} \quad f(x) &= -x^2 + 3 \quad \to \quad f'(x) = \frac{d}{dx}(-x^2) + \frac{d}{dx}(3) \\ +&= -\frac{d}{dx}(x^2) + \frac{d}{dx}(3) \\ +&= -2x^{2-1} + 0 \\ +&= -2x \\[10pt] +\text{(3)} \quad f(x) &= \frac{4x^3}{3} - x + 2e^x \quad \to \quad f'(x) = \frac{4}{3} \frac{d}{dx}(x^3) - \frac{d}{dx}(x) + 2 \frac{d}{dx}(e^x) \\ +&= \frac{4}{3}(3x^2) - 1 + 2e^x \\ +&= 4x^2 - 1 + 2e^x \\[10pt] +\text{(4)} \quad f(x) &= \frac{3}{x^2} + \sqrt{x} \quad \to \quad f'(x) = 3 \frac{d}{dx}(x^{-2}) + \frac{d}{dx}(x^{1/2}) \\ +&= 3(-2x^{-3}) + \frac{1}{2} x^{-1/2} \\ +&= \frac{-6}{x^3} + \frac{1}{2\sqrt{x}} \\[10pt] +\text{(5)} \quad f(x) &= \log(4x) \quad \to \quad f'(x) = \frac{d}{dx} [\log 4 + \log x] \\ +&= \frac{d}{dx}(\log 4) + \frac{d}{dx}(\log x) \\ +&= 0 + \frac{1}{x \ln 10} +\end{align*} $$ ## Product rule From f1ec1893d9613432b3366cec56c56efca518260e Mon Sep 17 00:00:00 2001 From: Michael Howard Date: Fri, 9 May 2025 13:46:19 -0500 Subject: [PATCH 5/5] Add simple derivative rules --- calculus/derivative-rules.md | 103 ++++++++++++++++++++++++----------- 1 file changed, 71 insertions(+), 32 deletions(-) diff --git a/calculus/derivative-rules.md b/calculus/derivative-rules.md index 67879a570..26bd70170 100644 --- a/calculus/derivative-rules.md +++ b/calculus/derivative-rules.md @@ -1,37 +1,76 @@ # Rules for derivatives -$$ -\begin{array}{lll} -\text{Rule} & f(x) & f'(x) \\ -\hline -\text{Constant} & k & 0 \\ -\text{Power} & x^n & nx^{n-1} \\ -\text{Multiple} & k u(x) & k u'(x) \\ -\text{Sum} & u(x) + v(x) & u'(x) + v'(x) \\ -\text{Difference} & u(x) - v(x) & u'(x) - v'(x) \\ -\text{Exponential} & a^x & \frac{a^x}{\ln a} \\ -\text{Logarithm} & \log_a x & \frac{1}{x \ln a} \\ -\end{array} -$$ - -$$ -\begin{align*} -\text{(1)} \quad f(x) &= x^3 \quad \to \quad f'(x) = 3x^{3-1} = 3x^2 \\[10pt] -\text{(2)} \quad f(x) &= -x^2 + 3 \quad \to \quad f'(x) = \frac{d}{dx}(-x^2) + \frac{d}{dx}(3) \\ -&= -\frac{d}{dx}(x^2) + \frac{d}{dx}(3) \\ -&= -2x^{2-1} + 0 \\ -&= -2x \\[10pt] -\text{(3)} \quad f(x) &= \frac{4x^3}{3} - x + 2e^x \quad \to \quad f'(x) = \frac{4}{3} \frac{d}{dx}(x^3) - \frac{d}{dx}(x) + 2 \frac{d}{dx}(e^x) \\ -&= \frac{4}{3}(3x^2) - 1 + 2e^x \\ -&= 4x^2 - 1 + 2e^x \\[10pt] -\text{(4)} \quad f(x) &= \frac{3}{x^2} + \sqrt{x} \quad \to \quad f'(x) = 3 \frac{d}{dx}(x^{-2}) + \frac{d}{dx}(x^{1/2}) \\ -&= 3(-2x^{-3}) + \frac{1}{2} x^{-1/2} \\ -&= \frac{-6}{x^3} + \frac{1}{2\sqrt{x}} \\[10pt] -\text{(5)} \quad f(x) &= \log(4x) \quad \to \quad f'(x) = \frac{d}{dx} [\log 4 + \log x] \\ -&= \frac{d}{dx}(\log 4) + \frac{d}{dx}(\log x) \\ -&= 0 + \frac{1}{x \ln 10} -\end{align*} -$$ +Some useful rules for derivatives to apply are: + +| Rule | $f(x)$ | $f'(x)$ | +|-------------|------------------|----------------------| +| Constant | $k$ | $0$ | +| Power | $x^n$, $n \ne 0$ | $nx^{n-1}$ | +| Multiple | $k u(x)$ | $k u'(x)$ | +| Sum | $u(x) + v(x)$ | $u'(x) + v'(x)$ | +| Difference | $u(x) - v(x)$ | $u'(x) - v'(x)$ | +| Exponential | $a^x$ | $a^x \ln a$ | +| Logarithm | $\log_a x$ | $\dfrac{1}{x \ln a}$ | + +Note that two special cases of the last two derivatives occur when $a = e$: + +\begin{equation} +\dd{}{e^x}{x} = e^x \qquad \dd{}{\ln x}{x} = \frac{1}{x} +\end{equation} + +1. $\displaystyle f(x) = x^3$ + + ```{solution} + \begin{equation} + f'(x) = 3x^{3-1} = 3x^2 + \end{equation} + ``` + +2. $\displaystyle f(x) = -x^2 + 3$ + + ```{solution} + \begin{align} + f'(x) &= \frac{d}{dx}(-x^2) + \frac{d}{dx}(3) \\ + &= -\frac{d}{dx}(x^2) + \frac{d}{dx}(3) \\ + &= -2x^{2-1} + 0 \\ + &= -2x + \end{align} + ``` + +3. $\displaystyle f(x) = \frac{4x^3}{3} - x + 2e^x$ + + ```{solution} + \begin{align} + f'(x) &= \frac{4}{3} \frac{d}{dx}(x^3) - \frac{d}{dx}(x) + + 2 \frac{d}{dx}(e^x) \\ + &= \frac{4}{3}(3x^2) - 1 + 2e^x \\ + &= 4x^2 - 1 + 2e^x + \end{align} + ``` + +4. $\displaystyle f(x) = \frac{3}{x^2} + \sqrt{x}$ + + ```{solution} + \begin{align} + f'(x) &= 3 \frac{d}{dx}(x^{-2}) + \frac{d}{dx}(x^{1/2}) \\ + &= 3(-2x^{-3}) + \frac{1}{2} x^{-1/2} \\ + &= \frac{-6}{x^3} + \frac{1}{2\sqrt{x}} + \end{align} + ``` + +5. $f(x) = \log(4x)$ + + ```{solution} + \begin{align} + f'(x) &= \frac{d}{dx} [\log 4 + \log x] \\ + &= \frac{d}{dx}(\log 4) + \frac{d}{dx}(\log x) \\ + &= 0 + \frac{1}{x \ln 10} + \end{align} + ``` + +There are other rules related to functions that are products, quotients, +compositions, or trigonometric. We will go over each of those in more detail +next. ## Product rule