Differentiation Rules

Differentiation Rules: Level 3 Challenges


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

For non-zero values of f(x)f(x), simplify the expression below. log2f(2016)(x)f(x)\log_{2} \dfrac{f^{(2016)} (x)}{f(x)}

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

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

Let f:RRf:\mathbb{R}\rightarrow \mathbb{R} be defined by f(x)=x3+3x+1f(x) = x^3+3x+1 and gg be the inverse of ff. If the value of g(5)g''(5) is equal to ab\dfrac{-a}{b}, where aa and bb are coprime positive integers, find the value of a+ba+b.

Find a closed-form expression for the following derivative.


F(x)=f(x)g(x)h(x){F(x)} = {f(x)g(x)h(x)}

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

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


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