Question

f(x) = integrate t ^ 4 dt from 3 to x

Answer 1

We have the function f(t) defined as:

$f(x)=\int_3^xt^4dt$

We have to find f'(x).

We can solve this integral as:

$f(x)=\frac{t^5}{5}|^x_3=\frac{x^5}{5}-\frac{3^5}{5}$

If we derive this expression for f(x) we obtain:

$f^{\prime}(x)=\frac{d}{dx}(\frac{x^5}{5}-\frac{3^5}{5})=\frac{5}{5}x^4+0=x^4$

NOTE: This expression could have been derived from the function inside the integral.

We can now find f'(3) as:

$f^{\prime}(3)=3^4=81$

Answer:

f'(x) = x^4

f'(3) = 81