Chain Rule
This rule states that:
CHAIN RULE OF DIFFERENTIATION:
Let's work some examples
Step 1: Identify the inner function and rewrite the outer function replacing the inner function by the variable u. |
g = 5x + 3 Inner Function u = 5x + 3 Set Inner Function to u f = u2 Outer Function |
Step 2: Take the derivative of both functions. |
Derivative of f = u2 Original Power & Constant
______________________________ Derivative of g = x + 3 Original Sum Rule Constant Multiple 5x0 + 0 Power & Constant 5 |
Step 3: Substitute the derivatives and the original expression for the variable u into the Chain Rule and simplify.
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2u(5) Chain Rule 2(5x + 3)(5) Substitute for u 50x + 30 Simplify |
If the expression is simplified first, the chain rule is not needed. |
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Step 1: Simplify |
(5x + 3)2 = (5x + 3)(5x + 3) 25x2 + 15x + 15x + 9 25x2 + 30x + 9 |
Step 2: Differentiate without the chain rule. |
Original Sum Rule Constant Multiple 25(2x1) + 30x0 + 0 Power & Constant 50x + 30 |
Step 1: Identify the inner function and rewrite the outer function replacing the inner function by the variable u. |
g = x2 - x + 14 Inner Function u = x2 - x + 14 Set IF to u Outer Function |
Step 2: Take the derivative of both functions. |
Derivative of Original Power
__________________________ Derivative of g = x2 - x + 14 Original Sum/Diff Rule 2x1 - 1x0 + 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 Sub
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Step 1: Identify the inner function and rewrite the outer function replacing the inner function by the variable u. |
Inner Function Set IF to u f = u2 Outer Function |
Step 2: Take the derivative of both functions. |
Derivative of f = u2 Original 2u1 Power
________________________ Derivative of Original Apply Chain Rule:
Take deriv. Simplify
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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 Sub for u
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Related Links: Math algebra Trigonometry Differentiation Rules Inverse Trigonometric Differentiation Rules Algebra Topics |
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