Question

The graph of the derivative of a linear function is
a parabola
a straight line parallel to the x-axis
a straight line parallel to the y-axis ............................

Answer 1

hold on.....
how to do dis type of quesn.......
\[x ^{x ^{}}\]

Answer 2

linear function defined as y = mx+b
_ > dy/dx = m
m is constant and graph of constant is a straight horizontal line

Answer 3

*...........\[x ^{x ^{x}}\]

Answer 4

thnx dumbcow......

Answer 5

\[\text{Let } y = x^x \]
\[\implies \ln y = x \ln x \]

And differentiate implicitly.

Same applies to higher powers, just repeat.

Answer 6

You could also write

\[x^x = e^{\ln x^x} = e^{x \ln x} \]

Answer 7

got it!!!!!!!!

Answer 8

how to do dis 1........\[x ^{^{x ^{x}}}\]

Answer 9

sorry but i m unable 2 understand

Answer 10

\[\text{Let } y = x^x \]
\[\implies \ln y = x \ln x \implies \frac{1}{y}\frac{\mathbb{d}y}{\mathbb{d}x} = \ln x + 1 \] \[\implies \frac{\mathbb{d}}{\mathbb{d}x} x^x = x^x(\ln x+1) \]
\[\text{Now let } z = x^{x^x} \]
\[\implies \ln z = x^x \ln x\ \implies \frac{1}{z}\frac{\mathbb{d}z}{\mathbb{d}x} = \frac{\mathbb{d}}{\mathbb{d}x} x^x \cdot \ln x + \frac{\mathbb{d}}{\mathbb{d}x} \ln x \cdot x^x \]

But from above we know \[ \frac{\mathbb{d}}{\mathbb{d}x} x^x = x^x(\ln x+1) \]

\[\implies \frac{1}{z}\frac{\mathbb{d}z}{\mathbb{d}x} = x^x \left( \ln x \cdot (\ln x + 1) + \frac{1}{x} \right) \]
\[\implies \frac{\mathbb{d}}{\mathbb{d}x} x^{x^x} = x^{x^x}\left[ x^x \left( \ln x \cdot (\ln x + 1) + \frac{1}{x} \right) \right] \]

Sorry for any typos.

Answer 11

Similarly

\[ \frac{\mathbb{d}}{\mathbb{d}x} x^{x^{x^{x}}} = x^{x^{x^{x}}} \left[ x^{x^x}\left[ \ln x \left( x^x \left( \ln x \cdot (\ln x + 1) + \frac{1}{x} \right)\right) + \frac{1}{x} \right] \right] \]

It is left to the readers to generalise this for
\[x^{x^{x^{x^{x^{x^{x^{.^{.^{.}}}}}}}}} \]