Science:Math Exam Resources/Courses/MATH101/April 2011/Question 01 (b)/Solution 1

Using Hint 2, we may immediately rewrite $f(t)$ as

$f(t)=-\int _{2}^{t^{4}}{\sqrt {1+x^{3}}}\,dx.$

We would like to use Hint 1, but it would only be applicable if the upper limit of integration were $t$, which in this case is not true. So we define

$g(u)=-\int _{2}^{u}{\sqrt {1+x^{3}}}\,dx$.

From Hint 1, we see immediately that $g'(u)=-{\sqrt {1+u^{3}}}$. We also see that $f(t)=g(t^{4})$. Using the chain rule, we can then compute that

$f'(t)=g'(t^{4})\cdot (t^{4})'=-{\sqrt {1+(t^{4})^{3}}}\cdot 4t^{3}=-4t^{3}{\sqrt {1+t^{12}}}.$