Find the solution of each of the following initial-value problems.

Chain Rule We need to find derivatives of compositions of 2 or more functions. Ex. f(x) = sin(x^2), g(x)= x + 1, h(x)= cos(sintan(x)) The chain rule F(x)=f(g(x)), F’(x)=f’(g(x))*g’(x) Ex. F(x) = sin(x^2) the other function f(g(x)) is sin(g(x)) the inner function g(x) is x^2 F’(x) = cos(x^2)*d/dx x^2,cos(x^2)*2x Ex. g(x) = x + 1, d/dx( x + 1) = d/dx (x^2+1)^½ The outer function is the power of ½, the inner function is x^2 +1 d/dx [ x^2 +1)^½] = ½(x^2+1)^-½* d/dx(x^2+1) = ½(X^2+1)^-½*2x this answer is perfectly acceptable Ex. d/dx [(x^4+x)^10] = 10(x^4+x)^9 * 4x^3 +1 Ex. d/dx (sin^8(x)) Remember sin^n(x) = (sin(x))^n So d/dx [(sin(x))^8] =8(sin(x))^7 * cos(x) Mini Formula = d/dx ((f(x)^n) = n/f(x))^n-1 * f’(x) Advice always rewrite trig functions like the equation above