Use limits to compute the derivative

Question

Use limits to compute the derivative.

 

f'(4) if f(x)=5x³

Answer 1

Definition of derivative at the point x₀: 

 

f'(x₀)=lim_h→0[f(x₀+h)-f(x₀)]/h

In your question x₀=4, so

f'(0)=lim_h→0[5(4+h)³-5*4³]/h

=5lim_h→0[(4+h)³-4³]/h =|here we use a³-b³=(a-b)(a²+ab+b²)

=5lim_h→0[(4+h-4)(16+4(4+h)+4²⁾]/h=

=5lim_h→0[h(16+16+4h+16)]/h=

=5lim_h→0[48+4h)]=5*48=240.

Thus, f'(4)=160.

 

Answer 2

f'(4) = lim_h→0 [(f(4+h) - f(4))/h]

 

        = lim_h→0 [(5(4+h)³ - 324)/h]

 

        = lim_h→0 [(5(64 + 48h + 12h² + h³ - 324)/h]

 

         = lim_h→0[ (240h + 60h² + 5h³)/h]

 

          = lim_h→0(240 + 60h + 5h²) = 240