Product, Quotient & Chain Rule – Practice ( Qn 1 – 5)

Question No: 1

Find the differentiation of \(f(x)=\frac{4}{5} x^5+\frac{3}{2} x^2-5 x+6\)
Answer : \( f^{\prime}(x)=4 x^4+3 x-5\)
. \( \begin{aligned} & f(x)=\frac{4}{5} x^5+\frac{3}{2} x^2-5 x+6 \\ & f^{\prime}(x)=\frac{4}{5} \times 5 x^4+\frac{3}{2} \times 2 x-5(1)+0 \\ & f^{\prime}(x)=4 x^4+3 x-5 \end{aligned}\) .

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Question No: 2

Use the product rule to find the differentiation of \(x^3 \ln x\)
Answer : \( x^2[1+3 \ln x]\)
. \( \begin{aligned} \frac{d}{d x}\left(x^3 \ln x\right) & =x^3(\ln x)^{\prime} +(\ln x)\left(x^3\right)^{\prime} \\ & =x^3 \times \frac{1}{x}+\ln x \times 3 x^2 \\ & =x^2+3 x^2 \ln x \\ & =x^2[1+3 \ln x] \end{aligned}\) ..

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Question No: 3

Use the quotient rule to find the differentiation of \(\frac{e^x}{x^3}\)
Answer : \(\frac{e^x[x-3]}{x^4}\)
. \( \begin{aligned} \frac{d}{d x}\left(\frac{e^x}{x^3}\right) & =\frac{x^3\left(e^x\right)-e^x\left(3 x^2\right)}{\left(x^3\right)^2} \\ & =\frac{x^3 e^x-3 x^2 e^x}{x^6} \\ & =\frac{x^2 e^x[x-3]}{x^6} \\ & =\frac{e^x[x-3]}{x^4} \end{aligned}\) .

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Question No: 4

Use the quotient rule to find the differentiation of \(\frac{\ln x}{x^2}\)
Answer : \( \frac{1-\ln x^2}{x^3}\)
. \( \begin{aligned} \frac{d}{d x}\left(\frac{\ln x}{x^2}\right) & =\frac{x^2(\ln x)^{\prime}-(\ln x)\left(x^2\right)^{\prime}}{\left(x^2\right)^2} \\ & =\frac{x^2 \times \frac{1}{x}-\ln x \times 2 x}{x^4} \\ & =\frac{x-2 x \ln x}{x^4} \\ & =\frac{x(1-2 \ln x)}{x^4} \\ & =\frac{1-\ln x^2}{x^3} \end{aligned}\) .

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Question No: 5

Use the quotient rule to find the differentiation of \(y=\frac{\sqrt{x}}{1-2 x}\)
Answer : \( \frac{1+2 x}{2 \sqrt{x}(1-2 x)^2}\)
. \( \begin{aligned} \frac{d y}{d x} & =\frac{(1-2 x) \times \frac{1}{2 \sqrt{x}}-\sqrt{x}(-2)}{(1-2 x)^2} \\ & =\frac{(1-2 x)+2 \sqrt{x}(2 \sqrt{x})}{2 \sqrt{x}} \\ & =\frac{1-2 x+4 x}{2 \sqrt{x}(1-2 x)^2} \\ & =\frac{1+2 x}{2 \sqrt{x}(1-2 x)^2} \end{aligned}\) .

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