Find the derivative of the function.
To find: The derivative of y=(x+1x)5.
The derivative of y=(x+1x)5 is y′=5(x2+1)4(x2−1)x6_.
The function is y=(x+1x)5.
The Chain Rule:
If h is differentiable at x and g is differentiable at h(x), then the composite function F=g∘h defined by F(x)=g(h(x)) is differentiable at x and F′ is given by the product
(1) Power Rule: ddx(xn)=nxn−1.
(2) Sum Rule: ddx[f(x)+g(x)]=ddx[f(x)]+ddx[g(x)].
Obtain the derivative of y.
Let h(x)=x+1x and g(u)=(u)5 where u=h(x).
Apply the chain rule as shown in equation (1),
The derivative g′(h(x)) is computed as follows,
Apply the power rule (1) and simplify further.
Thus, the derivative is g′(h(x))=5(x+1x)4 (3)
The derivative of h(x) is computed as follows,
Apply the sum rule (2) and the power rule (1),
Thus, the derivative is h′(x)=1−1x2 (4)
Substitute equations (3) and (4) in equation (2),
Therefore, the derivative of y=(x+1x)5 is 5(x2+1)4(x2−1)x6_.