Some Fundamental theory of calc questions

Question

I just wanted to sanity check these questions:

Find the derivative of these functions

1. $$g(s) = \int_{5}^s (t - t^2) ^8 dt$$

$$ g'(s) = (t - t^2)^8$$

2. $$h(x) = \int_{1}^\sqrt{x} \frac{z^2}{z^4 +1} dz$$

$$h'(x) = \frac{1}{2\sqrt{x}} \cdot \frac{z^2}{z^4 + 1}$$

Answer 1

Your first function $g'(s)$ still has $t$'s left over. I assume that once you replace them you will be OK.

Your second function $h'(x)$ still has $z$'s left over. I assume that once you replace them you will be OK.

Answer 2

$g'(s)$ means that g is a function of the variable s but you still have t which is only valid inside the integrand... $$g'(s)=(s-s^2)^8$$

Answer 3

For the first one just replace $t$ with $s$ in the integrand.

For the second I suppose the chain rule gives: $$h'(x)=\frac1{2\sqrt x}\cdot \frac x{x^2+1}$$.

Answer 4

The first should be $g'(s)=(s-s^2)^8$ and the second is $h'(x)=\frac1{2\sqrt x}\cdot \frac x{x^2+1}$

This follows from the Leibniz integral rule of differentiation under the integral sign , which is,

$\,\frac{d}{dx}\bigg(\huge \int_{\small a(x)}^{\small b(x)}\large f(x,t)\,dt\bigg) =\small f(x,b(x)).\frac{d(b(x))}{dx}-f(x,a(x)).\frac{d(a(x))}{dx}+\int_{\small{a(x)}}^{\small{b(x)}}\partial_xf(x,t)dt$.

In your case since the partial derivatives $\,0$ you can just omit them.

EDIT:

you are doing $\frac{d}{dx}(\int_1^{\sqrt x}\frac{z^2}{z^4+1})$. Thus you sub in $z=\sqrt x$ after you differentiate(because of the above mentioned rule) You get $\frac{(\sqrt x)^2}{(\sqrt x)^4+1}\cdot\frac1{2\sqrt x} = \frac{x}{x^2+1}\cdot\frac1{2\sqrt x}$. Also you dont get the term $\frac{1^2}{1^4+1}$ as $\frac{d}{dx}(1) =0$. Cancelling that term.