Derivatives: chain rule and other advanced topics

Chain rule

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Chain rule
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Common chain rule misunderstandings
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Chain rule
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Identifying composite functions
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Worked example: Derivative of cos³(x) using the chain rule
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Worked example: Derivative of √(3x²-x) using the chain rule
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Worked example: Derivative of ln(√x) using the chain rule
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Worked example: Chain rule with table
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Practice

Identify composite functionsGet 3 of 4 questions to level up!
Chain rule introGet 3 of 4 questions to level up!
Chain rule with tablesGet 3 of 4 questions to level up!

More chain rule practice

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Derivative of aˣ (for any positive base a)
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Derivative of logₐx (for any positive base a≠1)
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Worked example: Derivative of 7^(x²-x) using the chain rule
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Worked example: Derivative of log₄(x²+x) using the chain rule
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Worked example: Derivative of sec(3π/2-x) using the chain rule
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Worked example: Derivative of ∜(x³+4x²+7) using the chain rule
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Proving the chain rule
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Derivative rules review
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Practice

Derivatives of aˣ and logₐxGet 3 of 4 questions to level up!
Chain rule capstoneGet 3 of 4 questions to level up!

Implicit differentiation

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Implicit differentiation
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Worked example: Implicit differentiation
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Worked example: Evaluating derivative with implicit differentiation
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Showing explicit and implicit differentiation give same result
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Implicit differentiation review
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Practice

Implicit differentiationGet 3 of 4 questions to level up!

Implicit differentiation (advanced examples)

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Implicit differentiation (advanced example)
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Implicit differentiation (advanced example)
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Implicit differentiation (advanced example)
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Derivative of ln(x) from derivative of 𝑒ˣ and implicit differentiation
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Implicit differentiation review
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Differentiating inverse functions

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Derivatives of inverse functions
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Derivatives of inverse functions: from equation
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Derivatives of inverse functions: from table
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Practice

Derivatives of inverse functionsGet 3 of 4 questions to level up!

Derivatives of inverse trigonometric functions

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Derivative of inverse sine
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Derivative of inverse cosine
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Derivative of inverse tangent
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Differentiating inverse trig functions review
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Practice

Derivatives of inverse trigonometric functionsGet 3 of 4 questions to level up!

Strategy in differentiating functions

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Differentiating functions: Find the error
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Manipulating functions before differentiation
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Strategy in differentiating functions
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Practice

Differentiating functions: Find the errorGet 3 of 4 questions to level up!
Manipulating functions before differentiationGet 3 of 4 questions to level up!

Differentiation using multiple rules

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Differentiating using multiple rules: strategy
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Applying the chain rule and product rule
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Applying the chain rule twice
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Derivative of eᶜᵒˢˣ⋅cos(eˣ)
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Derivative of sin(ln(x²))
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Product rule to find derivative of product of three functions
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Practice

Differentiating using multiple rules: strategyGet 3 of 4 questions to level up!
Differentiating using multiple rulesGet 3 of 4 questions to level up!

Second derivatives

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Second derivatives
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Second derivatives (implicit equations): find expression
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Second derivatives (implicit equations): evaluate derivative
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Second derivatives review
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Practice

Second derivativesGet 3 of 4 questions to level up!
Second derivatives (implicit equations)Get 3 of 4 questions to level up!

Disguised derivatives

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Disguised derivatives
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Practice

Disguised derivativesGet 3 of 4 questions to level up!

Logarithmic differentiation

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Composite exponential function differentiation
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Worked example: Composite exponential function differentiation
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Practice

Composite exponential function differentiationGet 3 of 4 questions to level up!

Proof videos

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Proof: Differentiability implies continuity
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If function u is continuous at x, then Δu→0 as Δx→0
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Chain rule proof
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Quotient rule from product & chain rules
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About this unit

The chain rule tells us how to find the derivative of a composite function. This is an exceptionally useful rule, as it opens up a whole world of functions (and equations!) we can now differentiate. Also learn how to use all the different derivative rules together in a thoughtful and strategic manner.