I am struggling with this question;
I used the definition of $F'(x)$ and directional derivatives,$$F'(x)=\lim_{h\to0} \frac{F(x+h)-F(x)}{h}=\lim_{h\to0} \frac{E(x+h,x+h)-E(x,x)}{h}$$ $$=\sqrt{2}\lim_{s\to0} \frac{E(x+\frac{1}{\sqrt{2}}s,x+\frac{1}{\sqrt{2}}s)-E(x,x)}{s}$$$$ = \nabla E(x,y) \cdot (1,1) \bigg|_{y=x}$$ $$=\frac{\partial E}{\partial x}(x,y) \bigg|_{y=x}+\frac{\partial E}{\partial y}(x,y)\bigg|_{y=x}$$
but i was in trouble for solving $\frac{\partial E}{\partial x}(x,y) \bigg|_{y=x}$.
If I use Leibniz rule and taking x-partial derivative to $\sqrt{t^4+x^3}$ , then it is impossible to integrate with respect to $t$.
Please help me. thanks!
The question explicitly ask you to use chain rule to find $F'(x)$. Do you know what that means?
Do you know how to find the derivative in general for $$ G(x)=\int_{0}^{b(x)}f(x,t)\ dt, $$ or more general $$ G(x)=\int_{a(x)}^{b(x)}f(x,t)\ dt? $$ If you are still confused, take a look at
Differentiation under the integral sign.