Find the derivative of x^n-a^n x-a for some constant a. - Vidyadip

Question

Find the derivative of $\frac{\text{x}^\text{n}-\text{a}^\text{n}}{\text{x}-\text{a}}$ for some constant a.

✓

Answer

Let $\text{f}(\text{x})=\frac{\text{x}^\text{n}-\text{a}^\text{n}}{\text{x}-\text{a}}$ $\Rightarrow\text{f}'(\text{x})=\frac{\text{d}}{\text{dx}}\bigg(\frac{\text{x}^\text{n}-\text{a}^\text{n}}{\text{x}-\text{a}}\bigg)$ By quotient rule, $ \text{f}'(\text{x})=\frac{({\text{x}-\text{a})\frac{\text{d}}{\text{dx}}(\text{x}^\text{n}-\text{a}^\text{n})-(\text{x}^\text{n}-\text{a}^\text{n})}\frac{\text{d}}{\text{dx}}(\text{x}-\text{a})}{(\text{x}-\text{a})^2}$ $ =\frac{({\text{x}-\text{a})(\text{nx}^\text{n-1}-0)-(\text{x}^\text{n}-\text{a}^\text{n})}}{(\text{x}-\text{a})^2}$ $ =\frac{{\text{nx}^\text{n}-\text{anx}^\text{n-1}-\text{x}^\text{n}+\text{a}^\text{n}}}{(\text{x}-\text{a})^2}$

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 "(Each question 4 marks)" from LIMITS AND DERIVATIVES→LIMITS AND DERIVATIVES — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the angles between the following pairs of straight lines: 3x + 4y - 7 = 0 and 4x - 3y + 5 = 0

The vertices of a quadrilateral are A (-2, 6), B (1, 2), C (10, 4) and D (7, 8). Find the equation of its diagonals.

The diameters of circles (in mm) drawn in a design are given below:

Diameters	33-36	37-40	41-44	45-48	49-52
No. of circles	15	17	21	22	25

Calculate the standard deviation and mean diameter of the circles.
[Hint First make the data continuous by making the classes as 32.5-36.5, 36.5-40.5, 40.5-44.5, 44.5 - 48.5, 48.5 - 52.5 and then proceed.]

If $\text{f}(\text{x})=\begin{cases}a & x = 0\\b & x > 0\end{cases}\begin{cases}\text{mx}^2+\text{n,} & x < 0\\\text{nx}+m, & 0\leq\text{x}\leq 1\\\text{nx}^3+\text{x}&\text{x}>1\end{cases}$ . For what integers m and n does both $\lim\limits_{\text{x}\rightarrow0}\text{f}(\text{x})$ and $\lim\limits_{\text{x}\rightarrow1}\text{f}(\text{x})$exist?

Prove that the family of lines represented by $\text{x}(1+\lambda)=\text{y}(2-\lambda)+5=0,$ $\lambda$ being arbitrary, pass through a fixed point. Also, find that point.

Differentiate the following from the first principle$\text{x}\cos\text{x}$

Find the coordinates of the vertices of a triangle, the equations of whose sides are: x + y - 4 = 0, 2x - y + 3 = 0 and x - 3y + 2 = 0

Let r and n be positive integers such that 1 < r < n. Then prove the following: $\text{n}\ {{^\text{n-1}}\text{C}_{\text{r-1}}}=(\text{n}-\text{r}+1){{^\text{n}}\text{C}_{\text{r}-1}}$

Planes are drawn through the points (5, 0, 2) and (3, -2, 5) parallel to the coordinate planes. Find the lengths of the edges of the rectangular parallelopiped so formed.

Find sum of these numbers in G.P. is $21$ and the sum of their aquares is $189$. Find the numbers.