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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