Differentiate the following from first principle:e^ -x - Vidyadip

Question

Differentiate the following from first principle:$\text{e}^{-\text{x}}$

✓

Answer

We have, $\text{f(x)}=\text{e}^{-\text{x}}$ $\because\text{f}'\text{(x)}=\lim\limits_{\text{h}\rightarrow0}\frac{\text{f}\big(\text{x+h}\big)-\text{f}\big(\text{x}\big)}{\text{h}}$ $=\lim\limits_{\text{h}\rightarrow0}\frac{\text{e}^{-(\text{x+h})}-\text{e}^{-\text{x}}}{\text{h}}$ $=\lim\limits_{\text{h}\rightarrow0}\frac{\text{e}^{-\text{x}}\big(\text{e}^{-\text{h}}-1\big)}{\text{h}}$ $=\lim\limits_{\text{h}\rightarrow0}\frac{-\text{e}^{-\text{x}}\big(\text{e}^{-\text{h}}-1\big)}{-\text{h}}$ $= -\text{e}^{-\text{x}}$ $\Big[\because\lim\limits_{\theta\rightarrow0}\frac{\text{e}^{\theta}-1}{\theta}=1\Big]$

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 Derivatives→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 $\lim\limits_{\text{x}\rightarrow-\frac{5}{2}}{[\text{x}]}.$

Differentiate the following from first principle$\text{a}^{\sqrt{\text{x}}}$

The $4^{th}$ term of a G.P. is square of its second term, and the first firm is -3. Find its $7^{th}$ term.

For any two sets of A and B, prove that:

$\text{A}'\cup\text{B}=\text{U}\Rightarrow\text{A}\subset\text{B.}$

Prove that $\cos\frac{2\pi}{15}\cos\frac{4\pi}{15}\cos\frac{8\pi}{15}\cos\frac{16\pi}{15}=\frac{1}{16}$

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $(\text{ax+b})(\text{cx+d})^2$

Find the equations to the straight lines which pass through the origin and are inclined at an angle of 75° to the straight line $\text{x}+\text{y}+\sqrt{3}(\text{y}-\text{x})=\text{a}.$

Solve the following equations: $\sin\text{x}+\cos\text{x}=1$

Show that $\lim\limits_{\text{x}\rightarrow0}\ \sin\frac{1}{\text{x}}$ does not exist.

The equations of perpendicular bisectors of the sides AB and AC of a triangle ABC are $x - y + 5 = 0$ and $x + 2y = 0$ respectively. If the point A is $(1, -2)$, find the equation of the line BC.