Question

Use the rules of differentiation to differentiate the following functions.

1.f(x)=2x³+6x

2.g(x)=7x⁴-3x²

3.y(x)=(4x)³-18x²+6x

4.h(x)=(3x+4)²

5.h(x)=9x⅔+2/4√x

Answer 1

1.$f(x)=2x^3+6x$

here we apply the sum rule

=$d/dx(2x^3)+d/dx(6x)$

=$6x^2+6$

2.$g(x)=7x^4-3x^2$

here we apply the difference rule

$=d/dx(7x^4)-d/dx(3x^2)$

$=28x^3-6x$

3.$y(x)=(4x)^3-18x^2+6x$

$=$ $(4x^3)=64x^3$

$=64x^3-18x^2+6x$

we apply the sum/difference rule

$=d/dx(64x^3)-d/dx(18x^2)+d/dx(6x)$

$=192x^2-36x+6$

4.$h(x)=(3x+4)^2$

here we apply the chain rule

$d/dx[f(g(x)]=d/d[g(x)][f(x)]*d/dx(g(x))$

let $f(x)=2$

$g(x)=3x+4)$

$d/dx[(3x+4)^2]=2*(3x+4) *d/dx(3x+4)$

$d/dx(3x+4)= 3$

$=3(6x+8)$

$=18x+24$

5.$h(x)= 9x$^2/3 + $2/4\sqrt{ x}$

here we apply the sum rule

d/dx(9x^2/3) =6x^-1/3

to get the derivative of d/dx ($2/4 ​ \sqrt{x}$ )

= $2/4d/dx(1/ ​ \sqrt{x}$ )

$=2/4(-1/2(x^-1/2-1))$

$=-1/4x$^3/2

=6x^-1/3-1/4x^3/2