Logarithmic Differentiation
Before beginning our discussion, let's review the Laws of Logarithms.
LAWS OF LOGARITHMS:
If x and y are positive numbers, then
Law 1:
Law 2:
Law 3: If
Let's look at some examples:
Step 1: Take the natural log of both sides of the equation and use the law of logarithms to simplify. |
Take ln of both sides.
Apply Laws of Logarithms.
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Step 2: Differentiate implicitly. To differentiate the two terms on the right-hand side, apply the Chain Rule |
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Step 3: Solve for . Multiply both sides by y. |
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Step 4: Substitute for y.
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Step 1: Take the natural log of both sides of the equation and use the law of logarithms to simplify. |
Take ln Apply Law 1 of Logarithms
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Step 2: Differentiate implicitly. Apply the Product & Chain Rules to the right hand side. |
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Step 3: Solve for and simplify. Multiply both sides by y. |
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Step 4: Substitute for y.
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Step 1: Take the natural log of both sides of the equation and use the law of logarithms to simplify. Remember that . |
Take ln
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Step 2: Differentiate implicitly. |
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Step 3: Solve for . Multiply both sides by y. |
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Step 4: Substitute for y.
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This function can also be solved explicitly using the chain rule. |
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Step 1: Identify the inner function and rewrite the outer function replacing the inner function by the variable u. |
g = 5x2 - x + 4 Inner Function u = 5x2 - x + 4 f = eu Outer Function |
Step 2: Take the derivative of both functions. |
Derivative of f = eu Original eu Nat. Exp. Rule
__________________________ Derivative of g = 5x2 - x + 4 Original Sum/Diff Rule Constant Multiple 5(2x) - 1 + 0 Power & Constant
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Step 3: Substitute the derivatives and the original expression for the variable u into the Chain Rule and simplify.
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Chain Rule Substitute for u
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Related Links: Math algebra Implicit Differentiation General Differentiation Rules Algebra Topics |
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