using definition of a derivative, find the derivartive of : f(x)= x^3-7x^2+x+1

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justaguide | College Teacher | (Level 2) Distinguished Educator

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According to the definition of derivative f'(x) = lim h-->0[f(x+h)-f(h)]/h

Here f(x) = x^3 - 7x^2 + x + 1

f'(x) = lim h-->0[f(x + h) - f(x)]/h

=> f'(x) = lim h-->0[(x+h)^3 - 7(x+h)^2 + (x+h) + 1 - x^3 - 7x^2 - x - 1]/h

=> f'(x) = lim h-->0[x^3 + h^3 + 3h^2x + 3x^2h - 7x^2 - 7h^2 - 14xh + x + h + 1 - x^3 + 7x^2 - x - 1]/h

=> f'(x) = lim h-->0[x^3 - x^3 + h^3 + 3h^2x + 3x^2h - 7x^2 + 7x^2 - 7h^2 - 14xh + x - x + h + 1 - 1]/h

=> f'(x) = lim h-->0[ h^3 + 3h^2x + 3x^2h - 7h^2 - 14xh + h]/h

=> f'(x) = lim h-->0[ h^2 + 3hx + 3x^2 - 7h - 14x + 1]

substitute h = 0

=> 3x^2 - 14x + 1

This gives the derivative of x^3 - 7x^2 + x + 1 as 3x^2 - 14x + 1

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