11. Differentiation

iGCSE (2021 Edition)

Given that $y=u^3$`y`=`u`3 and $u=2x+3$`u`=2`x`+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`)=(5`x`3−4`x`2+3`x`−5)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`)=^{4}√2`x`2+2`x`+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(4`x`2−3`x`+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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Use the derivatives of the standard functions x^n (for any rational n), together with constant multiples, sums and composite functions.