# How do you use part one of the fundamental theorem of calculus to find the derivative of the function g(x)= intcos(t^10) dt from [cos(x) to 10x] ?

Refer to explanation

#### Explanation:

It is

g(x)=int_cosx^(10x) cos(t^10)dt=> (dg(x))/dx=cos((10x)^10)*(d(10x))/dx-cos((cosx)^10)*d(cosx)/dt= 10*cos((10x)^10)+sinx*cos((cosx)^10)

Generally if have a function with formula

$g \left(x\right) = {\int}_{g} {\left(x\right)}^{k \left(x\right)} f \left(t\right) \mathrm{dt}$ then its derivative is

$\frac{d \left(g \left(x\right)\right)}{\mathrm{dt}} = f \left(k \left(x\right)\right) \cdot \left(\frac{\mathrm{dk} \left(x\right)}{\mathrm{dt}}\right) - f \left(g \left(x\right)\right) \cdot \left(\frac{\mathrm{dg} \left(x\right)}{\mathrm{dt}}\right)$