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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.

# Chain Rule

What is the derivative of $$y=4\tan^{4} x?$$

$$f(x)$$ is a differentiable function such that $$f(2)=3$$ and $$f'(2)=3.$$ If $$g(x)=\left(x^2f(x)\right)^2,$$ what is the value of $$g'(2)?$$

What is the derivative of $$y=3 ^{x^ 6-4}?$$

Consider the function $$f(x)=\log_{8} \left[ (3x-1)^3 \right] .$$ What is the value of $$a$$ such that $f'(a)=\frac{9}{\ln 8}?$

If the instantaneous rate of change of $$y=\sin^{-1} (3x)$$ at $$x=\frac{1}{6}$$ can be expressed as $$\frac{18}{\sqrt{a}},$$ what is the value of $$a?$$

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