Calculus

# Differentiation Rules: Level 3 Challenges

$f(x) = e^x \cdot \sin x$

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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