vault backup: 2025-01-30 09:33:44

.obsidian/plugins/obsidian-git/data.json

@@ -0,0 +1,27 @@
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+  "syncMethod": "merge",
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+  "treeStructure": false,
+  "refreshSourceControl": true,
+  "basePath": "",
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education/math/MATH1210 (calc 1)/Derivatives.md

@@ -34,9 +34,10 @@ Given the equation $y = f(x)$, the following are all notations used to represent
 - Where a sharp turn takes place
 - If the slope of the tangent line is vertical
 
-# Higher Order Differentials
-- Take the differential of a differential
+# Higher Order Derivatives
+- Take the derivative of a derivative
 
+# Exponential Derivative Formula
 Using the definition of a derivative to determine the derivative of $f(x) = x^n$, where $n$ is any natural number.
 
 $$ f'(x) = \lim_{h \to 0} \dfrac{(x + h)^n - x^n}{h} $$
@@ -61,4 +62,8 @@ $$ \dfrac{(x + h)^n - x^n}{h} = \lim_{h \to 0} \dfrac{(x^n + nx^{n-1}h + P_{n3}x
 $x^n$ cancels out, and then $h$ can be factored out of the binomial series. 
 
 This leaves us with:
-$$ \lim_{h \to 0} nx^{n-1} + P_{n3} x^{} $$
+$$ \lim_{h \to 0} nx^{n-1} + P_{n3} x^{n-2}*0 \cdots v * 0 $$
+
+The zeros leave us with:
+
+$$ f(x) = n, \space $f'(x) = nx^{n-1} $$