Question

Find the derivative of the following functions: a. f(x) = (x^3 + 5)^1/4 - 15e^x^3 b. f(x) = (x - 3)^2 (x - 5)/(x - 4)^2(x^2 + 3)^5

Answer 1

Explanation:

Given function is

(a)F(x)=
`\left ( x^(3)+5\right )^(0.25)-15e^{x^(3)}`

`F^(')\left ( x\right )`=
`0.25\left ( x^(3)+5\right )^(-0.75)\frac{\mathrm{d} x^(3)}{\mathrm{d} x}-15e^{x^(3)}\frac{\mathrm{d} x^(3)}{\mathrm{d} x}`

`F^(')\left ( x\right )=0.25\left ( x^(3)+5\right )^(-0.75)* 3x^(2)-15e^{x^(3)}* \left ( 3x^(2)\right )`

(b)F(x)=
`(\left ( x-3\right )^2\left ( x-5\right ))/(\left ( x-4\right )^2\left ( x^(2)+3\right )^5)`

`F^(')\left ( x\right )`=
`(\left [ 2\left ( x-3\right )\right \left ( x-5\right )+\left ( x-3\right )^2]\left [ \left ( x-4\right )^2\left ( x^2+3\right )^5\right ]-\left [ 2\left ( x-4\right )^(3)\left ( x^2+3\right )^5+5\left ( x^2+3\right )^4\left ( 2x\right )\left ( x-4\right )^2\right ]\left [ \left ( x-3\right )^2\left ( x-5\right )\right ])/(\left [\left ( x-4\right )^2\left ( x^2+3\right )^5\right ]^2)`