How do you use the definition of a derivative to find the derivative of #f(x)=x^3-2x^2+5x-6#?

Question

William Abdalla · Accepted Answer

#f'(x)=3x^2-4x+5#

Here ,

#f(x)=x^3-2x^2+5x-6=>f(t)=t^3-2t^2+5t-6#

As we are aware,

#f'(x)=lim_(t tox)(f(t)-f(x))/(t-x) to"definition"#

Substitute values of #f(t) and f(x)#

#f'(x)=lim_(t tox)((t^3-2t^2+5t-6)-(x^3-2x^2+5x-6))/(t-x)#

#f'(x)=lim_(t tox)((t^3-x^3)-2(t^2-x^2)+5(t-x))/(t-x)#

#f'(x)=lim_(t tox)(color(red)(cancel((t-x))){(t^2+tx+x^2)-2(t+x)+5(1)})/color(red)cancel(t-x)#

#f'(x)=lim_(t tox){(t^2+tx+x^2)-2(t+x)+5(1)}#

#f'(x)=(x^2+x*x+x^2)-2(x+x)+5(1)#

#f'(x)=(3x^2)-2(2x)+5(1)#

#:.f'(x)=3x^2-4x+5#

Samantha Adkison · Answer

#f'(x) = 3 x^2 - 4 x + 5# #"The definition of the derivative of "f(x)" is : "# #f'(x) = lim_{h->0} ( f(x+h) - f(x) ) / h# #"So here we have "# #lim_{h->0} ( (x+h)^3 - 2 (x+h)^2 + 5 (x+h) - 6 - x^3 + 2 x^2 - 5 x + 6 ) / h# #= ((x^3 + 3 h x^2 + 3 h^2 x + h^3) - 2 (x^2 + 2 h x + h^2) + 5 (x + h) - x^3 + 2 x^2 - 5 x)/h# #= lim_{h->0} (3 h x^2 + 3 h^2 x + h^3 - 4 h x - 2 h^2 + 5 h)/h# #= lim_{h->0} 3 x^2 + 3 h x + h^2 - 4 x - 2 h + 5# #= lim_{h->0} 3 x^2 - 4 x + 5 + h^2 + 3 h x - 2 h# #= 3 x^2 - 4 x + 5 + lim_{h->0} h^2 + 3 h x - 2 h# #= 3 x^2 - 4 x + 5 + 0# #= 3 x^2 - 4 x + 5#

Emilia Aguilar · Answer

undefined To find the derivative of f(x) = x^3 - 2x^2 + 5x - 6 using the definition of a derivative, we can follow these steps:

1. Start with the definition of the derivative: f'(x) = lim(h->0) [f(x + h) - f(x)] / h.

2. Substitute the given function f(x) = x^3 - 2x^2 + 5x - 6 into the definition.

3. Expand the function f(x + h) by substituting x + h into the function: f(x + h) = (x + h)^3 - 2(x + h)^2 + 5(x + h) - 6.

4. Simplify the expanded expression of f(x + h).

5. Substitute the original function f(x) = x^3 - 2x^2 + 5x - 6 and the simplified expression of f(x + h) into the definition of the derivative.

6. Simplify the expression [f(x + h) - f(x)] / h.

7. Take the limit as h approaches 0 to find the derivative f'(x).

8. Simplify the resulting expression to obtain the derivative of f(x) = x^3 - 2x^2 + 5x - 6.

Note: The detailed calculations involved in this process can be quite lengthy and may require algebraic manipulations.