Related rates nearly always depend on the chain rule, so you might want to try filling up this pattern...

... where straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case x, because we're speaking of rates of change with respect to x), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

Here we're given a formula relating two variables, which is the top row...

So differentiate with respect to the inner function, and the inner function with respect to x...

Spoiler:

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Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!