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Open in new tab Q.01-04
Compute the derivative of 
by (a) multiplying and then differentiating and (b) using the product rule.
The product rule
![\[\frac{d}{d x}[f(x) g(x)]=f(x) \frac{d}{d x}[g(x)]+g(x) \frac{d}{d x}[f(x)]\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-90068dd8a49fa50bc3365b422286c4cc_l3.png "Rendered by QuickLaTeX.com")
![\[\frac{d}{d x}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x) \frac{d}{d x}[f(x)]-f(x) \frac{d}{d x}(g(x)]}{[g(x)]^2}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-d04773304c355657fb8827a30460265c_l3.png "Rendered by QuickLaTeX.com")
Solution:
1.(a )
![\[f(x)=(x+1)(2 x-1)\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-a00d103838f3185be70af1493720ae3b_l3.png "Rendered by QuickLaTeX.com")
![\[f(x)=2 x^2-x+2 x-1=2 x^2+x-1\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-0482c5de1db3556c02cb1f0864831fc1_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{aligned} & f^{\prime}(x)=\frac{d}{d x}\left[2 x^2+x-1\right]=2 \frac{d}{d x}\left[x^2\right]+\frac{d}{d x}[x]-\frac{d}{d x}[1] \\ \\& f^{\prime}(x)=2(2)(x)^{2-1}+1-0 \quad \therefore \quad \frac{d}{d x}\left[x^r\right]=rx^{r-1} \\ \\& f^{\prime}(x)=4 x+1 \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-25f8fd93bae1f323c92bc8dc58b16520_l3.png "Rendered by QuickLaTeX.com")
1(b)
Differentiating w.r.t (using product rule)
(using product rule) ![\[\begin{aligned} f^{\prime}(x) & =(x+1) \frac{d}{d x}(2 x-1)+(2 x-1) \frac{d}{d x}(x+1) \\ \\ & \left.=(x+1)\left[\frac{d}{d x}(2 x)-\frac{d}{d x}(1)\right]+(2 x-1)\left[\frac{d}{d x}[x]+\frac{d}{d x}(1)\right]\right] \\ \\ & =(x+1)[2(1)-0]+(2 x-1)[1+0] \\ \\ & =2 x+2+2 x-1=4 x+1 \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-ce52a5ae0477df72dfa44d5cbf077aad_l3.png "Rendered by QuickLaTeX.com")
2. (a)
![\[f(x)=\left(3 x^2-1\right)\left(x^2+2\right)\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-f4ac3967e722388a876c3fc04eaff1a7_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{aligned} & f(x)=3 x^4+6 x^2-x^2-2 \\ \\& f(x)=3 x^4+5 x^2-2 \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-23ec89b19612f130b95ec3036dd202c8_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{aligned} f^{\prime}(x)=\frac{d}{d x}\left[3 x^4+5 x^2-2\right]=3 \frac{d}{d x}\left[x^4\right]+5 \frac{d}{d x}\left[x^2\right]-\frac{d}{d x}[2] \\ \\f^{\prime}(x)=3(4) x^{4-1}+5(2) x^{2-1}-0 \quad \therefore \frac{d}{d x}\left[x^r\right]=rx^{r-1} \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-043057eef927c30d9103803f57cf14f5_l3.png "Rendered by QuickLaTeX.com")
![\[f^{\prime}(x)=12 x^3+10 x\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-38367f7ce9c905b6d904ad2ad99f7558_l3.png "Rendered by QuickLaTeX.com")
(b) (using product rule)
![\[\quad f(x)=\left(3 x^2-1\right)\left(x^2+2\right)\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-2c145f807750ef832b834c86a9d99e08_l3.png "Rendered by QuickLaTeX.com")
(using product rule) ![\[\begin{aligned} f^{\prime}(x) & =\frac{d}{d x}\left[3 x^2-1\right]\left[x^2+2\right] \\ \\ & =\left(3 x^2-1\right) \frac{d}{d x}\left[x^2+2\right]+\left[x^2+2\right] \frac{d}{d x}\left[3 x^2-1\right] \\ \\& =\left(3 x^2-1\right)(2 x+0)+\left(x^2+2\right)(6 x-0) \\ \\&=6 x^3-2 x+6 x^3+12 x \\ \\& f^{\prime}(x)=12 x^3+10 x . \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-729785c72cd510e5571440836086a790_l3.png "Rendered by QuickLaTeX.com")
3.(a)
![\[f(x)=\left(x^2+1\right)\left(x^2-1\right)=x^4-1\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-572e9d921f68d63bc27eb488c8c50b75_l3.png "Rendered by QuickLaTeX.com")
differentiating w.r.t
![\[\begin{aligned} & f^{\prime}(x)=\frac{d}{d x}\left[x^4-1\right] \\ \\ & =\frac{d}{d x}\left[x^4\right]-\frac{d}{d x}[1] \\ \\ & =4 x^{4-1}-0=4 x^3 \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-e4c96241899ca54012829fb717397e6f_l3.png "Rendered by QuickLaTeX.com")
(b)
![\[f(x)=\left(x^2+1\right)\left(x^2-1\right)\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-f2c80ebe367c7d508ed231e87a13f876_l3.png "Rendered by QuickLaTeX.com")
w.r.t (using product rule). ![\[\begin{aligned} & f^{\prime}(x) =\frac{d}{d x}\left[x^2+1\right]\left[x^2-1\right] \\ \\ & =\left[x^2+1\right] \frac{d}{d x}\left[x^2-1\right]+\left(x^2-1\right) \frac{d}{d x}\left[x^2+1\right] \\ \\ & \left.=\left(x^2+1\right)\left[\frac{d}{d x}\left(x^2\right)-\frac{d}{d x}(1)\right]+\left(x^2-1\right)\left[\frac{d}{d x} x^2\right)+\frac{d}{d x}(1)\right] \\ \\ & =\left(x^2+1\right)(2 x-0)+\left(x^2-1\right)(2 x+0) \\ \\ f^{\prime}(x) & =2 x^3+2 x+2 x^3-2 x=4 x^3 \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-1cf6204fbefbac81cfb2a4c35a569193_l3.png "Rendered by QuickLaTeX.com")
4. (a) .
![\[\begin{aligned} & f(x)=\left(x+1\right)\left(x^2-x+1\right) \\ \\ & f(x)=x^3-x^2+x+x^2-x+1 \\ \\& f(x)=x^3+1 \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-e515c9637709436a866578ae59bb992f_l3.png "Rendered by QuickLaTeX.com")
. ![\[\begin{aligned} & f^{\prime}(x)=\frac{d}{d x}\left[x^3+1\right]=\frac{d}{d x}\left[x^3\right]+\frac{d}{d x}[1] \\ \\& f^{\prime}(x)=3 x^2+0=3 x^2 \quad \therefore \frac{d}{d x}\left[x^r\right]=r x^{r-1} \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-4cbff5b5a973d94bfa35e84d6152de34_l3.png "Rendered by QuickLaTeX.com")
(b) w.r.t 
( using product rule)
![\[\begin{aligned} & f(x)=(x+1)\left(x^2-x+1\right) \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-e69354ae17827e97de2da3e9f99b01aa_l3.png "Rendered by QuickLaTeX.com")
w.r.t ( using product rule) ![\[f^{\prime}(x) =\frac{d}{d x}\left[(x+1)(x^2-x+1)\right]\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-052d268ca563f280bf51a4edfd1576d3_l3.png "Rendered by QuickLaTeX.com")
![\[f^{\prime}(x)=(x+1) \frac{d}{d x}\left(x^2-x+1\right)+\left(x^2-x+1\right) \frac{d}{d x}(x+1)\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-5345af3067b8804cbbfff3c9656078f6_l3.png "Rendered by QuickLaTeX.com")
![\[f^{\prime}(x)=(x+1)(2x-1)+(x^2-x+1)\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-2da69f2dd7143aad0520d046421b6148_l3.png "Rendered by QuickLaTeX.com")
![\[f^{\prime}(x)=2x^2+2x-x-1+x^2-x+1\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-aff2e2481d6c8d3155383aa276df2d15_l3.png "Rendered by QuickLaTeX.com")
![\[f^{\prime}(x)=3x^2\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-91e7eea8c13366a4fb0e67997567b499_l3.png "Rendered by QuickLaTeX.com")
Q.5 Find 
where
where ![\[f(x)=\left(3x^2+6\right)\left(2x-\frac{1}{4}\right)\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-018823e9ece721658fcb786ea51c9ff1_l3.png "Rendered by QuickLaTeX.com")
Solution: w.r.t. (using product rule)
w.r.t. (using product rule) ![\[\begin{aligned} f^{\prime} (x)=\frac{d}{d x}\left[(3x^2+6)(2x-\frac{1}{4})\right] \\ \\ f^{\prime} (x)=(3x^2+6)\frac{d}{d x}(2x-\frac{1}{4})+(2x-\frac{1}{4})\frac{d}{dx}(3x^2+6) \\ \\ f^{\prime} (x)=(3x^2+6)(2)+(2x-\frac{1}{4})(6x) \\ \\ f^{\prime} (x)=6x^2+12+12x^2-\frac{3}{2}x \\ \\ f^{\prime} (x)=18x^2-\frac{3}{2}x+12 \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-5366476a34a413f6386a3b96c5020ba1_l3.png "Rendered by QuickLaTeX.com")
Q.6 Find where
where ![\[f(x)=\left(2-x-3x^3\right)\left(7+x^5\right)\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-ba3a44c09bb50ec53e1885a943a9d83d_l3.png "Rendered by QuickLaTeX.com")
Solution:
differentiating w.r.t. (using product rule)
(using product rule) ![\[\begin{aligned} f^{\prime} (x)=\frac{d}{d x}\left[(2-x-3x^3)(7+x^5)\right] \\ \\ f^{\prime} (x)=(2-x-3x^3)\frac{d}{d x}(7+x^5)+(7+x^5)\frac{d}{dx}(2-x-3x^3) \\ \\f^{\prime} (x)=(2-x-3x^3)(5x^4)+(7+x^5)(-1-9x^2) \\ \\f^{\prime} (x)=10x^4-5x^5-15x^7-7-x^5-63x^2-9x^7 \\ \\f^{\prime} (x)=-24x^7-6x^5+10x^4-63x^2-7 \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-1543de32827ae0ed2bc6d438976e2a02_l3.png "Rendered by QuickLaTeX.com")