The derivative $\frac{d}{dx}\left(x^{\cos\left(2x\right)}\right)$ results in $\left(-2x\sin\left(2x\right)\ln\left(x\right)+\cos\left(2x\right)\right)x^{\left(\cos\left(2x\right)-1\right)}$
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The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable). Derivatives are a fundamental tool of calculus.