Determine a function $h$ $h$ such that $\frac{d}{d x} [\sqrt{h (x)}] = \frac{3 x^{2}}{2 \sqrt{x^{3} + 4}} ?$ $d/(dx)[sqrt(h(x))] = (3x^2)/(2sqrt(x^3+4))?$

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Answer 1 · 2017-03-19T18:07:06

$h (x) = x^{3} + 4$ $h(x) = x^3+4$

Given:

$\frac{d}{d x} [\sqrt{h (x)}] = \frac{3 x^{2}}{2 \sqrt{x^{3} + 4}}$ $d/(dx)[sqrt(h(x))] = (3x^2)/(2sqrt(x^3+4))$

Applying the chain rule to the left side:

$1/(2sqrt(h(x)))h'(x) = (3x^2)/(2sqrt(x^3+4))$

Matching the argument under the radical:

$h(x) = x^3+4$

Confirmed by $h'(x) = 3x^2$

Determine a function hhh such that d/(dx)[sqrt(h(x))] = (3x^2)/(2sqrt(x^3+4))?ddx[√h(x)]=3x22√x3+4?d/(dx)[sqrt(h(x))] = (3x^2)/(2sqrt(x^3+4))?