Product, Quotient & Chain Rule – Practice ( Qn 11 – 15)

Question No: 11

Find the differentiation of \( f(x)=\sqrt{3 x-5 x^2+4}\)
Answer : \( \frac{3-10 x}{2 \sqrt{3 x-5 x^2}+4}\)
. \( \begin{aligned} f(x) & =\sqrt{3 x-5 x^2+4} \\ f^{\prime}(x) & =\frac{1}{2 \sqrt{3 x-5 x^2+4}} \times (3-10 x) \\ & =\frac{3-10 x}{2 \sqrt{3 x-5 x^2}+4} \end{aligned}\) .

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Question No: 12

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

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Question No: 13

Use the quotient rule to find the differentiation of \(f(x)=\ln 3 x \cdot \sin 2 x\)
Answer : \(2 \ln 3 x \cdot \cos 2 x+x \sin 2 x\)
. \( \begin{aligned} f(x) & =\ln 3 x \cdot x \sin 2 x \\ f^{\prime}(x) & =(\ln 3 x)(\sin 2 x)^{\prime}+(\sin 2 x)(\ln 3 x)^{\prime} \\ & =\ln 3 x \cdot 2 \cdot \cos 2 x+\sin 2 x \times \frac{1}{3 x} \times 3 \\ & =2 \ln 3 x \cdot \cos 2 x+x \sin 2 x \end{aligned}\) .

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Question No: 14

Use the quotient rule to find the differentiation of \(f(x)=e^{x^2} \times \sin (3 x+4)\)
Answer : \( e^{x^2}[3 \cos (3 x+4)+2 x \sin (3 x+4)]\)
. \( \begin{aligned} f(x) & =e^{x^2} \times \sin (3 x+4) \\ f^{\prime}(x)= & e^{x^2} \times \cos (3 x+4) \times 3 +\sin (3 x+4) e^{x^2} \times 2 x \\ = & e^{x^2}[3 \cos (3 x+4)+2 x \sin (3 x+4)] \end{aligned}\) .

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Question No: 15

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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