Logical Structure of the Differentiation Rules
Logical Structure of the Differentiation Rules.
|Rule |How we know it works |
| | |
|If [pic], |The tangent to a straight line is the line itself. |
|then [pic] |or - Compute [pic]. |
| | |
|If [pic], |Compute [pic] or use the “zoom in ‘til linear” argument as for the chain |
|then [pic] |rule. |
| | |
|If [pic], |Compute [pic] or use the “zoom in ‘til linear” argument as for the chain |
|then [pic] |rule.. |
| | |
| |“Zoom in” argument: |
|If [pic], |1. A function and its tangent are nearly identical near the point of |
|then [pic] |tangency, xo. |
| |2. The tangent to g at xo and the tangent to f at g(xo) are linear |
| |functions whose composition is a linear function which is nearly |
| |identical to [pic] near xo. |
| |3. The slope of the composition of the tangent lines is the product of |
| |the slopes of the tangents. Hence: |
| |[pic]. |
| | |
|If [pic], |Write [pic] and take derivative of both sides and solve for [pic]which |
|then [pic]. |"pops up" when using the chain rule on the left side. From now on this |
| |will be called the inverse function idea. |
| | |
|If [pic], |Take derivative of both sides of |
|then [pic] |[pic] |
| |and solve for [pic]. |
| | |
|If [pic], |Write [pic] and use the product, chain, and [pic] rules. |
|then [pic] | |
| | |
|If [pic], |Use binomial theorem and compute [pic]. |
|then [pic]. |[One could also use the product rule n-1 times on [pic].] |
| | |
| [pic] |Write [pic] and take the derivative of both sides. Solve for [pic] which |
|then [pic]. |"pops up" when using the chain rule on the left side. |
| | |
|If [pic], |Compute [pic]. |
|then [pic] | |
| | |
|[pic] |Write [pic] and and use the chain, (positive) power, and [pic] rules. |
|then [pic]. | |
| | |
|If [pic], |Compute [pic] using sin(x+h)=sin(x)cos(h)+cos(x)sin(h) and |
| |[pic] and [pic] |
|then [pic] | |
| | |
|If [pic], |Use cos(x)=sin(π/2 – x) and |
| |sin(x)=cos(π/2 – x) |
|then [pic] | |
| | |
|All other trig. Functions |Write the function in terms of sine and cosine and apply rules we know. |
| | |
|If [pic], |Use the inverse function idea and draw a triangle with an angle whose sine|
|then [pic]. |is x to find cos(arcsin(x)). |
| | |
|If [pic], |Use the inverse function idea and draw a triangle with an angle whose |
|then [pic]. |tangent is x to find sec2(arctan(x)). |
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