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2.02 Composite functions and the chain rule

Interactive practice questions

Given that $y=u^3$y=u3 and $u=2x+3$u=2x+3, define $y$y in terms of $x$x. Leave your answer in factored form.

Easy
< 1min

Consider the function $f\left(x\right)=\left(5x^3-4x^2+3x-5\right)^7$f(x)=(5x34x2+3x5)7.

Redefine the function as composite functions $f\left(u\right)$f(u) and $u\left(x\right)$u(x), where $u\left(x\right)$u(x) is a polynomial.

Easy
1min

Consider the function $f\left(x\right)=\sqrt[4]{2x^2+2x+3}$f(x)=42x2+2x+3.

Redefine the function as composite functions $u\left(x\right)$u(x) and $f\left(u\right)$f(u), where $u\left(x\right)$u(x) is a polynomial.

Easy
< 1min

Consider the function $f\left(x\right)=\frac{1}{\left(4x^2-3x+5\right)^3}$f(x)=1(4x23x+5)3.

Redefine the function as composite functions $f\left(u\right)$f(u) and $u\left(x\right)$u(x), where $u\left(x\right)$u(x) is a polynomial.

Easy
< 1min
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Outcomes

ACMMM105

understand the notion of composition of functions and use the chain rule for determining the derivatives of composite functions

ACMMM106

apply the product, quotient and chain rule to differentiate functions such as xe^x, tan⁡x,1/x^n, x sin⁡x, e^(−x)sin⁡x and f(ax+b)

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