Download App

CONTINUITY AND DIFFERENTIABILITY — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
Using mathematical induction prove that $\frac{\text{d}}{\text{dx}}(\text{x}^\text{n})=\text{nx}^{\text{n-1}}$ for all positive.
✓

Answer

To prove: $\text{P(n)}:\frac{\text{d}}{\text{dx}}\text{(x}^\text{n})=\text{nx}^{\text{n}-1}$ for all positive integers n
For n = 1,
$\text{P}(1):\frac{\text{d}}{\text{dx}}\text{(x)}=1=1.\text{x}^{1-1}$
$\therefore\ $P(n) is true for n = 1
Let P(k) is true for some positive integer k.
That is, $\text{P(k)}:\frac{\text{d}}{\text{dx}}\text{(x}^\text{k})=\text{kx}^{\text{k}-1}$
It has to be proved that  P(k + 1) is also true.
Consider $\frac{\text{d}}{\text{dx}}(\text{x}^{\text{k}+1})=\frac{\text{d}}{\text{dx}}(\text{x}.\text{x}^\text{k})$
$=\text{x}^\text{k}.\frac{\text{d}}{\text{dx}}(\text{x})+\text{x}.\frac{\text{d}}{\text{dx}}(\text{x}^\text{k})$ $[\text{By applying product rule}]$
$=\text{x}^\text{k}.1+\text{x}.\text{k}.\text{x}^{\text{k}-1}$
$=\text{x}^\text{k}+\text{kx}^\text{k}$
$=(\text{k}+1).\text{x}^\text{k}$
$=(\text{k}+1).\text{x}^{\text{(k}+1)-1}$
Thus, P(k + 1) is true whenever P(k) is true.
Therefore, by the principle of mathematical lnduction, the statement P(n) is true for every positive integer n.
Hence, proved.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Check the commutativity and associativity of the following binary operations:
$'*'$ on $Q$ defined by $a * b = ab^2$ for all $a, b \in Q$.
→
If $\text{A}=\begin{bmatrix}\cos\alpha&\sin\alpha\\-\sin\alpha&\cos\alpha\end{bmatrix},$ then verify that $A^TA = I_2.$
→
The mean of a binomial distribution is 20 and the standard deviation 4. Calculate the parameters of the binomial distribution.
→
Evaluate the following integrals:
$\int\frac{(\text{x}+1)\text{e}^\text{x}}{\cos^2(\text{xe}^\text{x})}\text{ dx}$
→
Find the position vector of a point R which divides the line joining the two points P and Q with position vectors $\overrightarrow{\text{OP}}=2\vec{\text{a}}+\vec{\text{b}}\text{ and }\overrightarrow{\text{OQ}}=\vec{\text{a}}-2\vec{\text{b}}$, respectively in the ratio 1 : 2 internally and externally.
→
Matrices $A$ and $B$ will be inverse of each other only if
→
$\int\frac{1+\cos4\text{x}}{\cot\text{x}-\tan\text{x}}\text{dx}$
→
Give examples of two one$-$one functions $f_1$ and $f_2$ from $R$ to $R,$ such that $f_1 + f_2 : R \rightarrow R.$ defined by $(f_1 + f_2)(x) = f_1(x) + f_2(x)$ is not one$-$one.
→
Let R be a relation on the set A of ordered pair of integers defined by (x, y)R(u, v) if xv = yu. Show that R is an equivalence relation.
→
By using properties of determinants, show that:
$\begin{vmatrix}a-b-c&2a&2a\\2b&b-c-a&2b\\2c&2c&c-a-b\end{vmatrix}=(a+b+c)^3$​​
→