# real analysis – Show that \$G:[c,d]to mathbb R\$ defined by \$G(x)=int_0^{f(x)}F(t)dt, xin [c,d] \$ is differentiable and \$G'(x)=Fcirc f(x) cdot f'(x).\$

Let $$F$$ be continuous on $$(a,b).$$ let $$f:(c,d)to mathbb R$$ be differential function satisfy $$f((c,d))subseteq (a,b).$$ Show that $$G:(c,d)to mathbb R$$ defined by $$G(x)=int_0^{f(x)}F(t)dt, xin (c,d)$$ is differentiable and $$G'(x)=Fcirc f(x) cdot f'(x).$$

My attempt

$$lim_{hto 0} frac{ G(x+h)-G(x)}{h}=lim_{hto 0} frac{ int_0^{f(x+h)}F(t)dt-int_0^{f(x)}F(t)dt}{h}=lim_{hto 0} frac{ int_{f(x)}^{f(x+h)}F(t)dt}{h}$$
I am not able to go beyond this.