1 ---------- log(x + 1) 3
3^(1/log(x + 1))
Let .
Then, apply the chain rule. Multiply by :
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
Differentiate term by term:
Apply the power rule: goes to
The derivative of the constant is zero.
The result is:
The result of the chain rule is:
The result of the chain rule is:
The result of the chain rule is:
Now simplify:
The answer is:
1 ---------- log(x + 1) -3 *log(3) -------------------- 2 (x + 1)*log (x + 1)
1 ---------- log(1 + x) / 2 log(3) \ 3 *|1 + ---------- + -----------|*log(3) | log(1 + x) 2 | \ log (1 + x)/ ------------------------------------------------- 2 2 (1 + x) *log (1 + x)
1 ---------- / 2 \ log(1 + x) | 6 6 log (3) 3*log(3) 6*log(3) | -3 *|2 + ---------- + ----------- + ----------- + ----------- + -----------|*log(3) | log(1 + x) 2 4 2 3 | \ log (1 + x) log (1 + x) log (1 + x) log (1 + x)/ --------------------------------------------------------------------------------------------- 3 2 (1 + x) *log (1 + x)