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defon
2 years ago
9

43. Find an equation of the tangent line to the curve

="y=10^{x}" alt="y=10^{x}" align="absmiddle" class="latex-formula"> at the point (1,10)
Mathematics
1 answer:
astraxan [27]2 years ago
8 0

Answer:

23x-y=13

Step-by-step explanation:

<u>we </u><u>need </u><u>to </u><u>find </u><u>the </u><u>derivative </u><u>with </u><u>respect </u><u>to </u><u>x</u>

<u>In(</u>y) =In(10^x)

In(y)=xIn(10)

<em><u>now </u></em><em><u>we </u></em><em><u>have </u></em><em><u>to </u></em><em><u>take </u></em><em><u>the </u></em><em><u>derivative</u></em><em><u> </u></em><em><u>with</u></em><em><u> </u></em><em><u>respect</u></em><em><u> </u></em><em><u>to </u></em><em><u>x</u></em>

Therefore, y'=yIn(10)

where y'=10^xIn(10), Chain rule!

note: First derivative function always give us the gradient value when x value is substituted in

Therefore, at x=1, our gradient will be m= 10^1In(10)

Gradient m=23.0 to the nearest tenth

<u>Finding</u><u> </u><u>the </u><u>equation</u><u> </u><u>of </u><u>the </u><u>line</u>

<em>y-y1=</em><em>m(</em><em>x-x1)</em>

<em>y-10=</em><em>2</em><em>3</em><em>(</em><em>x-1</em><em>)</em>

<em>y-10=</em><em>2</em><em>3</em><em>x</em><em>-</em><em>2</em><em>3</em>

<em>y-23x=</em><em>-</em><em>2</em><em>3</em><em>+</em><em>1</em><em>0</em>

<em>y</em><em> </em><em>-</em><em> </em><em>23x</em><em> </em><em>=</em><em> </em><em>-</em><em>1</em><em>3</em>

<em>so,</em><em> </em><em>the </em><em>equation</em><em> </em><em>of </em><em>the </em><em>line </em><em>tangent</em><em> </em><em>line </em><em>is </em>

<em>2</em><em>3</em><em>x</em><em> </em><em>-</em><em> </em><em>y </em><em>=</em><em>1</em><em>3</em>

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