First apply the chain rule: with u = 2x+1, f(x) = u^5, so f'(x) = 5u^4 · u' = 5(2x+1)^4 · 2 = 10(2x+1)^4. Now differentiate f'(x) = 10(2x+1)^4 again using the chain rule: f''(x) = 10 · 4(2x+1)^3 · 2 = 80(2x+1)^3.

Second derivative via the chain rule

Let $f(x) = (2x+1)^5$ .

Find $f''(x)$ .

Hint

First find $f'(x)$ using the chain rule with outer function $u^5$ and inner function $u = 2x+1$ . Then differentiate $f'(x)$ again, applying the chain rule a second time.

Step-by-step solution

First apply the chain rule: with $u = 2x+1$ , $f(x) = u^5$ , so $f'(x) = 5u^4 \cdot u' = 5(2x+1)^4 \cdot 2 = 10(2x+1)^4$ .

Now differentiate $f'(x) = 10(2x+1)^4$ again using the chain rule: $f''(x) = 10 \cdot 4(2x+1)^3 \cdot 2 = 80(2x+1)^3$ .

Third derivative via the chain rule

Find $f'''(x)$ .

Hint

Differentiate $f''(x) = 80(2x+1)^3$ one more time using the chain rule.

Step-by-step solution

Starting from $f''(x) = 80(2x+1)^3$ , apply the chain rule once more: the outer function is $80u^3$ and the inner function is $u = 2x+1$ .

This gives $f'''(x) = 80 \cdot 3(2x+1)^2 \cdot 2 = 480(2x+1)^2$ .

Second derivative via the chain rule

Third derivative via the chain rule

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