We have the function f(t) defined as:
[tex]f(x)=\int_3^xt^4dt[/tex]
We have to find f'(x).
We can solve this integral as:
[tex]f(x)=\frac{t^5}{5}|^x_3=\frac{x^5}{5}-\frac{3^5}{5}[/tex]
If we derive this expression for f(x) we obtain:
[tex]f^{\prime}(x)=\frac{d}{dx}(\frac{x^5}{5}-\frac{3^5}{5})=\frac{5}{5}x^4+0=x^4[/tex]
NOTE: This expression could have been derived from the function inside the integral.
We can now find f'(3) as:
[tex]f^{\prime}(3)=3^4=81[/tex]
Answer:
f'(x) = x^4
f'(3) = 81
