Respuesta :
The derivative and domain of the function f(x) = x²-2x³ is :[tex](-\infty, \infty)[/tex].
What is derivative of the function (differentiation)?
The slope of a function's graph or, more precisely, the slope of the tangent line at a point can be used to interpret a function's derivative.
Its computation actually stems from the slope formula for a straight line, with the exception that curves require the employment of a limiting procedure.
Calculation for the derivative:
Step 1: Use the definition of the derivative. Remember that f(x+h) means plug (x+h) into everywhere there is an "x" in f(x).
[tex]\begin{aligned}f^{\prime}(x) &=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\&=\lim _{h \rightarrow 0} \frac{(x+h)^{2}-2(x+h)^{3}-\left(x^{2}-2 x^{3}\right)}{h} \\&=\lim _{h \rightarrow 0} \frac{\left(x^{2}+2 h x+h^{2}\right)-2(x+h)(x+h)^{2}-x^{2}+2 x^{3}}{h} \quad \text { cancel } x^{2}\end{aligned}[/tex]
[tex]\begin{aligned}&=\lim _{h \rightarrow 0} \frac{\left(2 h x+h^{2}\right)-2(x+h)\left(x^{2}+2 h x+h^{2}\right)+2 x^{3}}{h} \\&=\lim _{h \rightarrow 0} \frac{\left(2 h x+h^{2}\right)-2\left(x^{3}+2 h x^{2}+h^{2} x+h x^{2}+2 h^{2} x+h^{3}\right)+2 x^{3}}{h} \\&=\lim _{h \rightarrow 0} \frac{2 h x+h^{2}-2 x^{3}-4 h x^{2}-2 h^{2} x-2 h x^{2}-4 h^{2} x-2 h^{3}+2 x^{3}}{h}\end{aligned}[/tex]
[tex]=\lim _{h \rightarrow 0} \frac{2 h x+h^{2}-6 h x^{2}-6 h^{2} x-2 h^{3}}{h}[/tex]
Step 2: Cancel out a factor of h from each term in the numerator with the h in the denominator. Then direct substitute h=0.
[tex]\begin{aligned}&=\lim _{h \rightarrow 0}\left(2 x+h-6 x^{2}-6 h x-2 h^{2}\right) \\&=2 x+0-6 x^{2}-6(0) x-2(0)^{2} \\&=2 x-6 x^{2}\end{aligned}[/tex]
f and f' are polynomials, so their domains are all real numbers.
Therefore, for f(x) = x²-2x³ domain of f and f' :[tex](-\infty, \infty)[/tex].
′To know more about differentiation, here
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The complete question is -
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative f(x) = x²-2x³.
