Differentiate a) f(x) = (3 x^2 - 1)(x^2 - 4 x) b) R(x) = cos^2 (2 x) / sin x c) g(x) = sec x - 2...

Question:

Differentiate {eq}a) \displaystyle f(x) = (3 x^2 - 1)(x^2 - 4 x) \\ b) \displaystyle R(x) = \frac{\cos^2 (2 x) }{\sin x} \\ c) \displaystyle g(x) = \frac{\sec x - 2 }{ \tan x } {/eq}

Differentiation

This question is from the differentiation and we have to differentiate the given function. Will use these formula to solve these question.

{eq}\text{Product rule}\\ \Rightarrow \ \frac{d}{dx}(u\cdotp{v})=u\frac{d}{dx}v+v\frac{d}{dx}u\\ \Rightarrow \ \frac{d}{dx}(\frac{u}{v})=\frac{v\frac{d}{dx}u-u\frac{d}{dx}v}{v^{2}} {/eq}

Answer and Explanation:

{eq}a) \ f(x)=(3x^{2}-1)(x^{2}-4x)\\ \text{differentiate with respect to x}\\ \Rightarrow \ f'(x)=(3x^{2}-1)\frac{d}{dx}(x^{2}-4x)+(x^{2}-4x)\frac{d}{dx}(3x^{2}-1)\\ \Rightarrow \ f'(x)=(3x^{2}-1)(2x-4)+6x(x^{2}-4x)\\ \Rightarrow \ f'(x)=6x^{3}-12x^{2}-2x+4+6x^{3}-24x^{2}\\ \Rightarrow \ f'(x)=12x^{3}-36x^{2}-2x+4\\ b) \ R(x)=\frac{cos^{2}(2x)}{sin(x)}\\ \text{differentiate with respect to x}\\ \Rightarrow \ R'(x)=\frac{sin(x)\frac{d}{dx}cos^{2}(2x)-sin(x)cos^{2}(2x)\frac{d}{dx}}{sin^{2}(x)}\\ \Rightarrow \ R'(x)=\frac{-4cos(2x)sin(x)sin(2x)-cos(x)cos^{2}(2x)}{sin^{2}(x)}\\ \Rightarrow \ R'(x)=-\frac{4cos(2x)sin(x)sin(2x)+cos(x)cos^{2}(2x)}{sin^{2}(x)}\\ c) \ g(x)=\frac{sec(x)-2}{tan(x)}\\ \text{differentiate with respect to x}\\ \Rightarrow \ g'(x)=\frac{tan(x)\frac{d}{dx}(sec(x)-2)-(sec(x)-2)\frac{d}{dx}tan(x)}{tan^{2}(x)}\\ \Rightarrow \ g'(x)=\frac{tan^{2}(x)sec(x)-(sec(x)-2)sec^{2}(x)}{tan^{2}(x)}\\ {/eq}


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