Write the derivative of f(x) = |x|3 at x = 0. - Vidyadip

Question

Write the derivative of f(x) = |x|³ at x = 0.

✓

Answer

Given: $\text{f(x)}=|\text{x}^3|=\begin{cases}\text{x}^3,&\text{x}\geq0\\-\text{x}^3,&\text{x}<0\end{cases}$
(LHL at x = 0)
$\lim_\limits{\text{x}\rightarrow0^{-}}\frac{\text{f(x)}-\text{f}(0)}{\text{x}-0}$
$=\lim_\limits{\text{x}\rightarrow0}\frac{\text{f}(0-\text{h})-\text{f}(0)}{\text{x}}$
$=\lim_\limits{\text{x}\rightarrow0}\frac{\text{h}^3}{-\text{h}}$
$=0$
(RHL at x = 0)
$\lim_\limits{\text{x}\rightarrow0^{+}}\frac{\text{f(x)}-\text{f}(0)}{\text{x}-0}$
$=\lim_\limits{\text{x}\rightarrow0}\frac{\text{f}(0+\text{h})-\text{f}(0)}{\text{x}}$
$=\lim_\limits{\text{x}\rightarrow0}\frac{\text{h}^3-0}{-\text{h}}$
$=0$
And f(0) = 0.
Thus, (LHL at x = 0) = (RHL at x = 0) = f(0)
Hence, $\lim_\limits{\text{x}\rightarrow0}\frac{\text{f(x)}-\text{f}(0)}{\text{x}-0}=\text{f}'(0)=0$.

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" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Show that the lines $\frac{\text{x}-5}{7}=\frac{\text{y}+2}{-5}=\frac{\text{z}}{1}$ and $\frac{\text{x}}{1}=\frac{\text{y}}{2}=\frac{\text{z}}{3}$ are perpendicular to each

Using properties of determinants, prove that:
$\begin{vmatrix}\sin\alpha&\cos\alpha&\cos(\alpha+\delta)\\\sin\beta&\cos\beta&\cos(\beta+\delta)\\\sin\gamma&\cos\gamma&\cos(\gamma+\delta)\end{vmatrix}=0$

Let * be a binary operation on Q₀ (set of non-zero rational numbers) defined by $\text{a}\ ^* \ \text{b}=\frac{\text{ab}}{5}$ for all $\text{a, b}\in\text{Q}_0.$ Show that * is commutative as well as associative. Also, find its identity element if it exists.

Evaluate the following definite integrals:
$\int_{0}^\limits{1}\text{x}(1-\text{x})^5\text{ dx}$

Find the integrals of the functions in Exercises:
$\cos2\text{x}\cos4\text{x}\cos6\text{x}$

Let +₆ (addition modulo 6) be a binary operation on S = {0, 1, 2, 3, 4, 5}. Write the value of 2 + ₆4⁻¹+ ₆3⁻¹.

$\text{If x}=\text{a}(\cos\text{t}+\text{t}\sin\text{t})\text{ and y}=\text{a}(\sin\text{t}-\text{t}\cos\text{t}),\ \text{find}\dfrac{\text{d}^2\text{y}}{\text{dx}^2}$

Evaluate the following definite integrals:
$\int_{0}^\limits{2}\frac{1}{\sqrt{3+2\text{x}-\text{x}^2}}\text{ dx}$

Evaluate the following integrals:

$\int\frac{\sin\text{x}}{\sqrt{4\cos^2\text{x}-1}}\text{ dx}$

Find the angle between the pair of lines
$\vec{r}=2 \hat{i}-5 \hat{j}+\hat{k}+\lambda(3 \hat{i}+2 \hat{j}+6 \hat{k})$ and $\vec{r}=7 \hat{i}-6 \hat{k}+\mu(\hat{i}+2 \hat{j}+2 \hat{k})$