Teil 1

Using the chain rule, ´d/dx(sin(1/(e^x (x^4+1)))) = ( dsin(u))/( du) ( du)/( dx)´, where ´u = e^(-x)/(x^4+1)´ and ´( d)/( du)(sin(u)) = cos(u)´: ´f'(x) = cos(1/(e^x (1+x^4))) d/dx(1/(e^x (1+x^4)))´

Use the quotient rule, ´d/dx(u/v) = (v ( du)/( dx)-u ( dv)/( dx))/v^2´, where ´u = e^(-x)´ and ´v = x^4+1´: ´f'(x) = ((x^4+1) d/dx(e^(-x))-(d/dx(1+x^4))/e^x)/((x^4+1)^2) cos(1/(e^x (1+x^4)))´

Using the chain rule, ´d/dx(e^(-x)) = (de^u)/(du) (du)/(dx)´, where ´u = -x´ and ´(d)/(du)(e^u) = e^u´: ´f'(x) = (cos(1/(e^x (1+x^4))) (-(d/dx(1+x^4))/e^x+(1+x^4) (d/dx(-x))/e^x)) / (1+x^4)^2´

Factor out constants: ´f'(x) = (cos(1/(e^x (1+x^4))) (-(d/dx(1+x^4))/e^x+((1+x^4) -d/dx(x))/e^x)) / (1+x^4)^2´

The derivative of ´x´ is ´1´: ´f'(x) = (cos(1/(e^x (1+x^4))) (-(d/dx(1+x^4))/e^x-(1 (1+x^4))/e^x))/(1+x^4)^2´

Differentiate the sum term by term: ´f'(x) = (cos(1/(e^x (1+x^4))) (-(1+x^4)/e^x-d/dx(1)+d/dx(x^4)/e^x))/(1+x^4)^2´

The derivative of 1 is zero: ´f'(x) = (cos(1/(e^x (1+x^4))) (-(1+x^4)/e^x-(d/dx(x^4)+0)/e^x))/(1+x^4)^2´

Simplify the expression: ´f'(x) = (cos(1/(e^x (1+x^4))) (-(1+x^4)/e^x-(d/dx(x^4))/e^x))/(1+x^4)^2´

Use the power rule, ´d/dx(x^n) = n x^(n-1)´, where ´n = 4: d/dx(x^4) = 4 x^3:´ ´f'(x) = (cos(1/(e^x (1+x^4))) (-(1+x^4)/e^x-4 x^3/e^x))/(1+x^4)^2´

Simplify the expression: ´f'(x) = ((-(4 x^3)/e^x-(1+x^4)/e^x) cos(1/(e^x (1+x^4))))/(1+x^4)^2´

Teil 2

´f'(x) = d/dx(sin^(-1)(x))´ ´f'(x) = 1/sqrt(1-x^2)´

Achtung: Nur bei passender Definitionsmenge und Wertebereich. Es müssen beide bijektiv sein:

arcsin: ´[-1,1] => [-pi/2, pi/2]´ sin: ´[-pi/2, pi/2] => [-1,1]´

Teil 3

´d/dx(sin) = d/dx(1/2 (e^x-e^(-x)))´

Factor out constants: ´cosh(x) = 1/2 (d/dx(-e^(-x)+e^x))´

Differentiate the sum term by term and factor out constants: ´cosh(x) = 1/2d/dx(e^x)-d/dx(e^(-x))´

Using the chain rule, ´d/dx(e^(-x)) = (de^u)/(du) (du)/(dx)´, where ´u = -x´ and ´(d)/(du)(e^u) = e^u´: ´cosh(x) = 1/2(d/dx(e^x)-(d/dx(-x))/e^x)´

Factor out constants: ´cosh(x) = 1/2 (d/dx(e^x)--d/dx(x)/e^x)´

Simplify the expression: ´cosh(x) = 1/2 (d/dx(e^x)+(d/dx(x))/e^x)´

The derivative of ´x´ is ´1´: ´cosh(x) = 1/2 (d/dx(e^x)+1/e^x)´

The derivative of ´e^x´ is ´e^x´: ´cosh(x) = 1/2 (e^(-x)+e^x)´

The same with ´cosh(x)´

´=> d/dx(cosh(x)) = d/dx(1/2(e^x + e^(-x)))´ ´sinh(x) = 1/2 (e^x - e^(-x))´