The derivative $\frac{d}{dx}\left(\sin\left(x+1\right)^{\left(x-1\right)}\right)$ results in $\left(\ln\left(\sin\left(x+1\right)\right)+\frac{\left(x-1\right)\cos\left(x+1\right)}{\sin\left(x+1\right)}\right)\sin\left(x+1\right)^{\left(x-1\right)}$
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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.