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Differentiation Rules

These are the rules that explain how to take derivatives of any functions: from polynomials to trigonometric functions to logarithms.

Level 3

         

\[f(x) = e^x \cdot \sin x \]

For non-zero values of \(f(x)\), simplify the expression below. \[\log_{2} \dfrac{f^{(2016)} (x)}{f(x)}\]

Notation: \(f^{(n)}(x) \) denotes the \(n^\text{th} \) derivative of \(f(x) \).

\[\large { \left. \frac { d }{ d(\cos { (x) } ) } \cos { (2015x) } \right\vert _{ x=2\pi } } = \ ? \]

Let \(f:\mathbb{R}\rightarrow \mathbb{R}\) be defined by \(f(x) = x^3+3x+1\) and \(g\) be the inverse of \(f\). If the value of \(g''(5) \) is equal to \(\dfrac{-a}{b}\), where \(a\) and \(b\) are coprime positive integers, find the value of \(a+b\).

Find a closed-form expression for the following derivative.

\[\frac{d}{d(x^{n})}x^{y}\]

\[{F(x)} = {f(x)g(x)h(x)}\]

The above equation is true for all real \(x\), where \(f(x)\), \(g(x)\) and \(h(x)\) are differentiable functions at some point \(a\).

Given \(F '(a) = 21 F(a), f '(a) = 4f(a), g'(a) = -7g(a), h'(a) = kh(a) \). Find \(k\).

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