practice exercises 19 - 60. derivatives find and simplify the derivative of the following functions. 19…

practice exercises 19 - 60. derivatives find and simplify the derivative of the following functions. 19. $f(x)=3x^{4}(2x^{2}-1)$ 20. $g(x)=6x - 2xe^{x}$ 21. $f(x)=\frac{x}{x + 1}$ 22. $f(x)=\frac{x^{3}-4x^{2}+x}{x - 2}$ 23. $f(t)=t^{5/3}e^{t}$ 24. $g(w)=e^{w}(5w^{2}+3w + 1)$ 25. $f(x)=\frac{e^{x}}{e^{x}+1}$ 26. $f(x)=\frac{2e^{x}-1}{2e^{x}+1}$ 27. $f(x)=xe^{-x}$ 28. $f(x)=e^{x}sqrt3{x}$

practice exercises 19 - 60. derivatives find and simplify the derivative of the following functions. 19. $f(x)=3x^{4}(2x^{2}-1)$ 20. $g(x)=6x - 2xe^{x}$ 21. $f(x)=\frac{x}{x + 1}$ 22. $f(x)=\frac{x^{3}-4x^{2}+x}{x - 2}$ 23. $f(t)=t^{5/3}e^{t}$ 24. $g(w)=e^{w}(5w^{2}+3w + 1)$ 25. $f(x)=\frac{e^{x}}{e^{x}+1}$ 26. $f(x)=\frac{2e^{x}-1}{2e^{x}+1}$ 27. $f(x)=xe^{-x}$ 28. $f(x)=e^{x}sqrt3{x}$

Answer

Explanation:

Step1: For $f(x)=3x^{4}(2x^{2}-1)$, use the product - rule $(uv)^\prime = u^\prime v+uv^\prime$. Let $u = 3x^{4}$ and $v=2x^{2}-1$.

$u^\prime=12x^{3}$, $v^\prime = 4x$. Then $f^\prime(x)=u^\prime v+uv^\prime=12x^{3}(2x^{2}-1)+3x^{4}(4x)$.

Step2: Expand the expression.

$f^\prime(x)=24x^{5}-12x^{3}+12x^{5}$.

Step3: Combine like - terms.

$f^\prime(x)=(24x^{5}+12x^{5})-12x^{3}=36x^{5}-12x^{3}$.

Answer:

$f^\prime(x)=36x^{5}-12x^{3}$