Derivative question

Question

I need to find the derivative of:

$$ h(x) = \int_{0}^{x^2} (1-t^2)^{1/3} \, dt $$

Would the answer to that just be:

$$ (1-x^4)^{1/3}? $$

$h(x) = g(x^2)$ where $g(x) = \int_0^x (1 - t^2)^{1/3} , dt$. What you've written down is $g'(x^2)$, which is not the same as $h'(x)$. Use the chain rule. — Qiaochu Yuan, May 16 '11 at 20:56
See also: http://math.stackexchange.com/questions/6155/derivative-of-integral/6156#6156 — Isaac, May 16 '11 at 22:53

score 5 · Answer 1 · answered May 16 '11 at 20:47

Let $v = x^2$ then $$\frac{\mathrm{d}}{\mathrm{d}v} \int_0^{v} f(t) \mathrm{d}t = f(v)$$ so $$\frac{\mathrm{d}}{\mathrm{d}x} \int_0^{x^2} f(t) \mathrm{d}t = \frac{\mathrm{d}v}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}v} \int_0^{v} f(t) \mathrm{d}t = 2x f(x^2).$$

score 3 · Answer 2 · edited Apr 13 '17 at 12:21

If $F(x)=\int_{a}^{x}f(t)\;\mathrm{d}t$, then $F^{\prime }(x)=f(x)$. By the chain rule if $F(x)=\int_{a}^{u(x)}f(t)\;\mathrm{d}t$, then

$$F^{\prime }(x)=F^{\prime }(u)u^{\prime }(x)=f(u(x))u^{\prime }(x).$$

In the present case $F(x)=h(x)$, $u(x)=x^{2}$ and $f(t)=(1-t^{2})^{1/3}$. Hence $u^{\prime }(x)=2x$ and $f(u(x))=f(x^{2})=(1-x^{4})^{1/3}$.

Thus

$$h^{\prime }(x)=2(1-x^{4})^{1/3}x.$$

A generalization is to find the derivative of an integral where both limits are functions of $x$, as in this question.

Derivative question

2 Answers2