Find the derivative of the following functions
Question: Find the derivative of the following functions (it is to be understood thata,b,c,d,p, q,randsare fixed non-zero constants andmandnare integers):$\frac{a+b \sin x}{c+d \cos x}$ Solution: Let $f(x)=\frac{a+b \sin x}{c+d \cos x}$ By quotient rule, $f^{\prime}(x)=\frac{(c+d \cos x) \frac{d}{d x}(a+b \sin x)-(a+b \sin x) \frac{d}{d x}(c+d \cos x)}{(c+d \cos x)^{2}}$ $=\frac{(c+d \cos x)(b \cos x)-(a+b \sin x)(-d \sin x)}{(c+d \cos x)^{2}}$ $=\frac{c b \cos x+b d \cos ^{2} x+a d \sin x+b d \...
Read More →Find the derivative of the following functions
Question: Find the derivative of the following functions (it is to be understood thata,b,c,d,p, q,randsare fixed non-zero constants andmandnare integers):$\sin ^{n} x$ Solution: Let $y=\sin ^{n} x$. Accordingly, for $n=1, y=\sin x$. $\therefore \frac{d y}{d x}=\cos x$, i.e., $\frac{d}{d x} \sin x=\cos x$ For $n=2, y=\sin ^{2} x$ $\therefore \frac{d y}{d x}=\frac{d}{d x}(\sin x \sin x)$ $=(\sin x)^{\prime} \sin x+\sin x(\sin x)^{\prime}$ [By Leibnitz product rule] $=\cos x \sin x+\sin x \cos x$ $...
Read More →Assuming that x, y, z are positive real numbers,
Question: Assuming that $x, y, z$ are positive real numbers, simplify each of the following: (i) $\left(\sqrt{x^{-3}}\right)^{5}$ (ii) $\sqrt{x^{3} y^{-2}}$ (iii) $\left(x^{-2 / 3} y^{-1 / 2}\right)^{2}$ (iv) $(\sqrt{x})^{-2 / 3} \sqrt{y^{4}} \div \sqrt{x y^{-1 / 2}}$ (v) $\sqrt[5]{243 x^{10} y^{5} z^{10}}$ (vi) $\left(\frac{x^{-4}}{y^{-10}}\right)^{5 / 4}$ (vii) $\left(\frac{\sqrt{2}}{\sqrt{3}}\right)^{5}\left(\frac{6}{7}\right)^{2}$ Solution: We have to simplify the following, assuming that $x...
Read More →Find the slope of the normal to the curve
Question: Find the slope of the normal to the curve $x=a \cos ^{3} \theta, y=a \sin ^{3} \theta$ at $\theta=\frac{\pi}{4}$. Solution: It is given that $x=a \cos ^{3} \theta$ and $y=a \sin ^{3} \theta$. $\therefore \frac{d x}{d \theta}=3 a \cos ^{2} \theta(-\sin \theta)=-3 a \cos ^{2} \theta \sin \theta$ $\frac{d y}{d \theta}=3 a \sin ^{2} \theta(\cos \theta)$ $\therefore \frac{d y}{d x}=\frac{\left(\frac{d y}{d \theta}\right)}{\left(\frac{d x}{d \theta}\right)}=\frac{3 a \sin ^{2} \theta \cos \t...
Read More →Find the derivative of the following functions
Question: Find the derivative of the following functions (it is to be understood thata,b,c,d,p, q,randsare fixed non-zero constants andmandnare integers):$\frac{\sec x-1}{\sec x+1}$ Solution: Let $f(x)=\frac{\sec x-1}{\sec x+1}$ $f(x)=\frac{\frac{1}{\cos x}-1}{\frac{1}{\cos x}+1}=\frac{1-\cos x}{1+\cos x}$ By quotient rule, $f^{\prime}(x)=\frac{(1+\cos x) \frac{d}{d x}(1-\cos x)-(1-\cos x) \frac{d}{d x}(1+\cos x)}{(1+\cos x)^{2}}$ $=\frac{(1+\cos x)(\sin x)-(1-\cos x)(-\sin x)}{(1+\cos x)^{2}}$ ...
Read More →Find the slope of the tangent to the curve
Question: Find the slope of the tangent to the curve $y=x^{3}-3 x+2$ at the point whose $x$-coordinate is $3 .$ Solution: The given curve is $y=x^{3}-3 x+2$. $\therefore \frac{d y}{d x}=3 x^{2}-3$ The slope of the tangent to a curve at $\left(x_{0}, y_{0}\right)$ is $\left.\frac{d y}{d x}\right\rfloor_{\left(x_{0}, y_{0}\right)}$. Hence, the slope of the tangent at the point where thex-coordinate is 3 is given by, $\left.\left.\frac{d y}{d x}\right]_{x=3}=3 x^{2}-3\right]_{x=3}=3(3)^{2}-3=27-3=2...
Read More →Find the slope of the tangent to curve
Question: Find the slope of the tangent to curve $y=x^{3}-x+1$ at the point whose $x$-coordinate is 2 . Solution: The given curve is $y=x^{3}-x+1$. $\therefore \frac{d y}{d x}=3 x^{2}-1$ The slope of the tangent to a curve at $\left(x_{0}, y_{0}\right)$ is $\left.\frac{d y}{d x}\right]_{\left(x_{0}, y_{0}\right)}$. It is given that $x_{0}=2$. Hence, the slope of the tangent at the point where thex-coordinate is 2 is given by, $\left.\left.\frac{d y}{d x}\right]_{x=2}=3 x^{2}-1\right]_{x=2}=3(2)^...
Read More →Find the slope of the tangent to the curve
Question: Find the slope of the tangent to the curve $y=\frac{x-1}{x-2}, x \neq 2$ at $x=10$. Solution: The given curve is $y=\frac{x-1}{x-2}$. $\therefore \frac{d y}{d x}=\frac{(x-2)(1)-(x-1)(1)}{(x-2)^{2}}$ $=\frac{x-2-x+1}{(x-2)^{2}}=\frac{-1}{(x-2)^{2}}$ Thus, the slope of the tangent atx= 10 is given by, $\left.\left.\frac{d y}{d x}\right]_{x=10}=\frac{-1}{(x-2)^{2}}\right]_{x=10}=\frac{-1}{(10-2)^{2}}=\frac{-1}{64}$ Hence, the slope of the tangent at $x=10$ is $\frac{-1}{64}$....
Read More →Find the derivative of the following functions
Question: Find the derivative of the following functions (it is to be understood thata,b,c,d,p, q,randsare fixed non-zero constants andmandnare integers):$\frac{\sin x+\cos x}{\sin x-\cos x}$ Solution: Let $f(x)=\frac{\sin x+\cos x}{\sin x-\cos x}$ By quotient rule, $f^{\prime}(x)=\frac{(\sin x-\cos x) \frac{d}{d x}(\sin x+\cos x)-(\sin x+\cos x) \frac{d}{d x}(\sin x-\cos x)}{(\sin x-\cos x)^{2}}$ $=\frac{(\sin x-\cos x)(\cos x-\sin x)-(\sin x+\cos x)(\cos x+\sin x)}{(\sin x-\cos x)^{2}}$ $=\fra...
Read More →If a=xyp−1,b=xyq−1 and c=xyr−1,
Question: If $a=x y^{p-1}, b=x y^{q-1}$ and $c=x y^{r-1}$, prove that $a^{q-r} b^{r-p} c^{p-q}=1$ Solution: It is given that $a=x y^{p-1}, b=x y^{q-1}$ and $c=x y^{r-1}$. $\therefore a^{q-r} b^{r-p} c^{p-q}$ $=\left(x y^{p-1}\right)^{q-r}\left(x y^{q-1}\right)^{r-p}\left(x y^{r-1}\right)^{p-q}$ $=x^{(q-r)} u^{(p-1)(q-r)} x^{(r-p)} u^{(r-p)(q-1)} x^{(p-q)} u^{(p-q)(r-1)}$ $=x^{(q-r)} x^{(r-p)} x^{(p-q)} y^{(p-1)(q-r)} y^{(r-p)(q-1)} y^{(p-q)(r-1)}$ $=x^{(q-r)+(r-p)+(p-q)} y^{(p-1)(q-r)+(r-p)(q-1)...
Read More →Find the slope of the tangent to the curve
Question: Find the slope of the tangent to the curve $y=3 x^{4}-4 x$ at $x=4$. Solution: The given curve is $y=3 x^{4}-4 x$. Then, the slope of the tangent to the given curve atx= 4 is given by, $\left.\left.\frac{d y}{d x}\right]_{x=4}=12 x^{3}-4\right]_{x=4}=12(4)^{3}-4=12(64)-4=764$...
Read More →Find the derivative of the following functions
Question: Find the derivative of the following functions (it is to be understood thata,b,c,d,p, q,randsare fixed non-zero constants andmandnare integers):$\frac{\cos x}{1+\sin x}$ Solution: Let $f(x)=\frac{\cos x}{1+\sin x}$ By quotient rule, $f^{\prime}(x)=\frac{(1+\sin x) \frac{d}{d x}(\cos x)-(\cos x) \frac{d}{d x}(1+\sin x)}{(1+\sin x)^{2}}$ $=\frac{(1+\sin x)(-\sin x)-(\cos x)(\cos x)}{(1+\sin x)^{2}}$ $=\frac{-\sin x-\sin ^{2} x-\cos ^{2} x}{(1+\sin x)^{2}}$ $=\frac{-\sin x-\left(\sin ^{2}...
Read More →Given 4725=3a5b7c, find
Question: Given $4725=3^{a} 5^{b} 7^{c}$, find (i) the integral values of $a, b$ and $c$ (ii) the value of $2^{-a} 3^{b} 7^{c}$ Solution: (i) Given $4725=3^{a} 5^{b} 7^{c}$ First find out the prime factorisation of 4725. It can be observed that 4725 can be written as $3^{3} \times 5^{2} \times 7^{1}$ $\therefore 4725=3^{a} 5^{b} 7^{c}=3^{3} 5^{2} 7^{1}$ Hence, $a=3, b=2$ and $c=1$ (ii) When $a=3, b=2$ and $c=1$, $2^{-a} 3^{b} 7^{c}$ $=2^{-3} \times 3^{2} \times 7^{1}$ $=\frac{1}{8} \times 9 \tim...
Read More →Find the derivative of the following functions
Question: Find the derivative of the following functions (it is to be understood thata,b,c,d,p, q,randsare fixed non-zero constants andmandnare integers): cosecxcotx Solution: Let $f(x)=\operatorname{cosec} x \cot x$ By Leibnitz product rule, $f^{\prime}(x)=\operatorname{cosec} x(\cot x)^{\prime}+\cot x(\operatorname{cosec} x)^{\prime}$$\ldots(1)$ Let $f_{1}(x)=\cot x$. Accordingly, $f_{1}(x+h)=\cot (x+h)$ By first principle, $f_{1}^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f_{1}(x+h)-f_{1}(x)}{...
Read More →The interval in which
Question: The interval in which $y=x^{2} e^{-x}$ is increasing is (A) $(-\infty, \infty)$ (B) $(-2,0)$ (C) $(2, \infty)$ (D) $(0,2)$ Solution: We have, $y=x^{2} e^{-x}$ $\therefore \frac{d y}{d x}=2 x e^{-x}-x^{2} e^{-x}=x e^{-x}(2-x)$ Now, $\frac{d y}{d x}=0$ $\Rightarrow x=0$ and $x=2$ The points $x=0$ and $x=2$ divide the real line into three disjoint intervals i.e., $(-\infty, 0),(0,2)$, and $(2, \infty)$. In intervals $(-\infty, 0)$ and $(2, \infty), f^{\prime}(x)0$ as $e^{-x}$ is always po...
Read More →Prove that the function given by
Question: Prove that the function given by $f(x)=x^{3}-3 x^{2}+3 x-100$ is increasing in $\mathbf{R}$. Solution: We have, $\begin{aligned} f(x) =x^{3}-3 x^{2}+3 x-100 \\ f^{\prime}(x) =3 x^{2}-6 x+3 \\ =3\left(x^{2}-2 x+1\right) \\ =3(x-1)^{2} \end{aligned}$ For any $x \in \mathbf{R},(x-1)^{2}0$ Thus, $f^{\prime}(x)$ is always positive in $\mathbf{R}$. Hence, the given function (f)is increasing inR....
Read More →Find the derivative of the following functions
Question: Find the derivative of the following functions (it is to be understood thata,b,c,d,p, q,randsare fixed non-zero constants andmandnare integers):$\sin (x+a)$ Solution: Let $f(x)=\sin (x+a)$ $f(x+h)=\sin (x+h+a)$ By first principle, $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ $=\lim _{h \rightarrow 0} \frac{\sin (x+h+a)-\sin (x+a)}{h}$ $=\lim _{h \rightarrow 0} \frac{1}{h}\left[2 \cos \left(\frac{x+h+a+x+a}{2}\right) \sin \left(\frac{x+h+a-x-a}{2}\right)\right]$ $=\lim ...
Read More →If 1176=2a3b7c, find a, b and c.
Question: If1176=2a3b7c1176=2a3b7c, finda,bandc. Solution: First find out the prime factorisation of 1176. It can be observed that 1176 can be written as $2^{3} \times 3^{1} \times 7^{2}$. $1176=2^{3} 3^{1} 7^{2}=2^{a} 3^{b} 7^{c}$ Hence, $a=3, b=1$ and $c=2$....
Read More →Prove that the function
Question: Prove that the function $f$ given by $f(x)=\log \cos x$ is strictly decreasing on $\left(0, \frac{\pi}{2}\right)$ and strictly increasing on $\left(\frac{\pi}{2}, \pi\right)$. Solution: We have, $f(x)=\log \cos x$ $\therefore f^{\prime}(x)=\frac{1}{\cos x}(-\sin x)=-\tan x$ In interval $\left(0, \frac{\pi}{2}\right), \tan x0 \Rightarrow-\tan x0$ $\therefore f^{\prime}(x)0$ on $\left(0, \frac{\pi}{2}\right)$ $\therefore f$ is strictly decreasing on $\left(0, \frac{\pi}{2}\right)$. In in...
Read More →Find the derivative of the following functions
Question: Find the derivative of the following functions (it is to be understood thata,b,c,d,p, q,randsare fixed non-zero constants andmandnare integers):$(a x+b)^{n}(c x+d)^{m}$ Solution: Let $f(x)=(a x+b)^{n}(c x+d)^{m}$ By Leibnitz product rule, $f^{\prime}(x)=(a x+b)^{\prime} \frac{d}{d x}(c x+d)^{m}+(c x+d)^{m} \frac{d}{d x}(a x+b)^{n}$$\ldots(1)$ Now, let $f_{1}(x)=(c x+d)^{m}$ $f_{1}(x+h)=(c x+c h+d)^{m}$ $f_{1}^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f_{1}(x+h)-f_{1}(x)}{h}$ $=\lim _{h...
Read More →Prove that the function
Question: Prove that the function $f$ given by $f(x)=\log \sin x$ is strictly increasing on $\left(0, \frac{\pi}{2}\right)$ and strictly decreasing on $\left(\frac{\pi}{2}, \pi\right)$. Solution: We have, $f(x)=\log \sin x$ $\therefore f^{\prime}(x)=\frac{1}{\sin x} \cos x=\cot x$ In interval $\left(0, \frac{\pi}{2}\right), f^{\prime}(x)=\cot x0$. $\therefore f$ is strictly increasing in $\left(0, \frac{\pi}{2}\right)$. In interval $\left(\frac{\pi}{2}, \pi\right), f^{\prime}(x)=\cot x0$ $\there...
Read More →If 49392=a4b2c3, find the values of a, b and c,
Question: If $49392=a^{4} b^{2} a^{2} c^{3}$, find the values of $a, b$ and $c$, where $a, b$ and $c$ are different positive primes. Solution: First find out the prime factorisation of 49392. It can be observed that 49392 can be written as $2^{4} \times 3^{2} \times 7^{3}$, where 2,3 and 7 are positive primes. $\therefore 49392=2^{4} 3^{2} 7^{3}=a^{4} b^{2} c^{3}$ $\Rightarrow a=2, b=3, c=7$...
Read More →Let I be any interval disjoint from (−1, 1).
Question: Let $I$ be any interval disjoint from $(-1,1)$. Prove that the function $f$ given by $f(x)=x+\frac{1}{x}$ is strictly increasing on $\mathrm{I} .$ Solution: We have, $f(x)=x+\frac{1}{x}$ $\therefore f^{\prime}(x)=1-\frac{1}{x^{2}}$ Now, $f^{\prime}(x)=0 \Rightarrow \frac{1}{x^{2}}=1 \Rightarrow x=\pm 1$ The points $x=1$ and $x=-1$ divide the real line in three disjoint intervals i.e., $(-\infty,-1),(-1,1)$, and $(1, \infty)$. In interval (1, 1), it is observed that: $-1x1$ $\Rightarrow...
Read More →Find the derivative of the following functions
Question: Find the derivative of the following functions (it is to be understood thata,b,c,d,p, q,randsare fixed non-zero constants andmandnare integers):$(a x+b)^{n}$ Solution: Let $f(x)=(a x+b)^{n}$. Accordingly, $f(x+h)=\{a(x+h)+b\}^{n}=(a x+a h+b)^{n}$ By first principle, $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f^{\prime}(x)}{h}$ $=\lim _{h \rightarrow 0} \frac{(a x+a h+b)^{n}-(a x+b)^{n}}{h}$ $=\lim _{h \rightarrow 0} \frac{(a x+b)^{n}\left(1+\frac{a h}{a x+b}\right)^{n}-(a x+b)...
Read More →Find the derivative of the following functions
Question: Find the derivative of the following functions (it is to be understood thata,b,c,d,p, q,randsare fixed non-zero constants andmandnare integers):$4 \sqrt{x}-2$ Solution: Let $f(x)=4 \sqrt{x}-2$ $f^{\prime}(x)=\frac{d}{d x}(4 \sqrt{x}-2)=\frac{d}{d x}(4 \sqrt{x})-\frac{d}{d x}(2)$ $=4 \frac{d}{d x}\left(x^{\frac{1}{2}}\right)-0=4\left(\frac{1}{2} x^{\frac{1}{2}-1}\right)$ $=\left(2 x^{-\frac{1}{2}}\right)=\frac{2}{\sqrt{x}}$...
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