The Chain Rule

Using the chain rule, differentiate y=sin(5x)

D=derivative. Recall that D sin(x)=cos(x) First differentiate the outer layer (the sine function), then multiply it by the derivative of the inner layer (5x): = D sin(5x)=cos(5x) * D (5x) =cos(5x)*5 =5cos(5x)

Using the chain rule, differentiate y=sin^3(x)

D=derivative. Recall that D sin(x)=cos(x), and sin^3(x)=(sin(x))^3 First differentiate the outer layer (the 3rd power), then multiply it by the derivative of the inner layer (the sine function): = D sin^3(x)=D (sin(x))^3=3(sin(x))^(3-1) * D sin(x) =3(sin(x))^2(cos(x)) =3(sin^2(x))(cos(x))

Using the chain rule, differentiate y= (3x+1)^2

D=derivative First differentiate the outer layer (the square), then multiply it by the derivative of the inner layer (3x+1): = D (3x+1)^2=2(3x+1)^(2-1) * D (3x+1) =2(3x+1)*(3) =6(3x+1) =18x+6

Using the chain rule, differentiate y=(1-8z)^(1/3)

D=derivative. First differentiate the outer layer (the 1/3 power), then multiply it by the derivative of the inner layer (1-8z): = D (1-8z)^(1/3)=(1/3)(1-8z)^(-2/3) * D (1-8z) =(1/3)(1-8z)^(-2/3)(-8) =(-8/3)(1-8z)^(-2/3)

Using the chain rule, differentiate y=(1-4x+7x^5)^30

D=derivative. First differentiate the outer layer (the 30th power), then multiply it by the derivative of the inner layer (1-4x+7x^5): = D (1-4x+7x^5)^30=30(1-4x+7x^5)^(30-1) * D (1-4x+7x^5) =30(1-4x+7x^5)^29 * (-4+35x^4) =30(35(x^4)-4)(1-4x+7x^5)^29

Using the chain rule, differentiate y=e^x^2

D=derivative. Recall that D e^x=e^x First differentiate the outer layer (e^x), then multiply it by the derivative of the inner layer (x^2): = D e^x^2= e^x^2 * D x^2 =2x(e^x^2)

Using the chain rule, differentiate y=ln(17-x)

D=derivative. Recall that D ln(x) = 1/x First differentiate the outer layer (the ln function), then multiply it by the derivative of the inner layer (17-x): = D ln(17-x)=(1/(17-x)) * D (17-x) =(1/(17-x))(-1) =(1/(x-17))