Find the derivative d/dx of the integral from x to a of cos(sin(t)) dt

To find the derivative of the integral, we can use the Fundamental Theorem of Calculus. According to this theorem, if we have an integral of the form ∫[a(x)]^[b(x)] f(t) dt, where both the lower limit a(x) and the upper limit b(x) depend on x, then we can find its derivative by evaluating the integrand f(t) at the upper limit b(x) and multiplying by the derivative of the upper limit b'(x), subtracting the value of the integrand at the lower limit a(x) multiplied by the derivative of the lower limit a'(x). Symbolically, this can be written as:

d/dx [∫[a(x)]^[b(x)] f(t) dt] = f(b(x)) * b'(x) - f(a(x)) * a'(x)

Applying this formula to our specific problem, we have:

d/dx [∫[x]^[a] cos(sin(t)) dt] = cos(sin(a)) * d/dx(a) - cos(sin(x)) * d/dx(x)

Since x is the independent variable, its derivative is 1:

d/dx [∫[x]^[a] cos(sin(t)) dt] = cos(sin(a)) * 0 - cos(sin(x)) * 1

Finally, simplifying the expression:

d/dx [∫[x]^[a] cos(sin(t)) dt] = -cos(sin(x))