Find the derivative of the function.

Question

Find the derivative of the function.

f(x) = ln( square root of (x) +7)



Answer 1

For real x > -7, write ln(square root of (x +7)) as ln[(x+7)^0.5].

Write the formula: d[ln u]/dx = (1/u) • (du/dx) for real u > 0.

For u equal to (x+7)^0.5, take du/dx as 0.5(1)(x+7)^-0.5.

Then (1/u) • (du/dx) = 1/(x+7)^0.5 • 0.5(1)(x+7)^-0.5

which reduces to 0.5/(x+7)^0.5+0.5 which simplifies to

0.5/(x+7)¹ or 1/2(x + 7) or 1/(2x + 14).

Note that ln (x^0.5 + 7) for positive x would be 0.5x^-0.5 +0 divided

by (x^0.5 + 7), which would simplify to 0.5/x^0.5/(x^0.5 + 7) equal to

1/[2x + 14x^0.5].