Using the chain rule,
Way 2: First, rewrite the function:
Then taking the derivative only requires the rule for constant multiples, therefore
For the pair of functions, determine what the chain rule says.
The input to the inside function, z, will occur only in denominators:
and the outermost variable, p, will occur only in numerators:
The chain rule says
The innermost variable is t:
and the outermost variable is q:
x = f(y) = f(g(z)).
The inner function is y, the innermost variable is z, and the outermost variable is x. The chain rule says
the chain rule says
The derivative of u with respect to s is es and the derivative of s with respect to x is -6x, therefore we find
The derivative of s with respect to t is 4t3 and the derivative of t with respect to r is 4r3, so we find
The derivative of z with respect to m is and the derivative of m with respect to n is cos n. Putting it all together gives us
The derivative of y with respect to z is 3z2 + 5z4 and the derivative of z with respect to x is 4x3 + 4, so
The derivative of r with respect to q is and the derivative of q with respect to p is
sec2 p, therefore
Rewrite y to remove the fraction.
Make it rain.
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