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Derivatives: chain rule and other advanced topics

Chain rule

Learn

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Common chain rule misunderstandings

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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

Learn

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)

Learn

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

Learn

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

Learn

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

Learn

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

Learn

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

Learn

Disguised derivatives

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Practice

Disguised derivativesGet 3 of 4 questions to level up!

Logarithmic differentiation

Learn

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

Learn

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.