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How to use it?
- Derivative of:
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Derivative of 4/x
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Derivative of 1/2*x^2
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Derivative of e^(5*x)
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Derivative of (x^2+1)^2
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Identical expressions
- y=sqrt((-x^ two +7x- ten)/(x^ two -7x+ twelve))
- y equally square root of (( minus x squared plus 7x minus 10) divide by (x squared minus 7x plus 12))
- y equally square root of (( minus x to the power of two plus 7x minus ten) divide by (x to the power of two minus 7x plus twelve))
- y=√((-x^2+7x-10)/(x^2-7x+12))
- y=sqrt((-x2+7x-10)/(x2-7x+12))
- y=sqrt-x2+7x-10/x2-7x+12
- y=sqrt((-x²+7x-10)/(x²-7x+12))
- y=sqrt((-x to the power of 2+7x-10)/(x to the power of 2-7x+12))
- y=sqrt-x^2+7x-10/x^2-7x+12
- y=sqrt((-x^2+7x-10) divide by (x^2-7x+12))
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Similar expressions
- y=sqrt((-x^2+7x+10)/(x^2-7x+12))
- y=sqrt((x^2+7x-10)/(x^2-7x+12))
- y=sqrt((-x^2+7x-10)/(x^2+7x+12))
- y=sqrt((-x^2+7x-10)/(x^2-7x-12))
- y=sqrt((-x^2-7x-10)/(x^2-7x+12))
Derivative of y=sqrt((-x^2+7x-10)/(x^2-7x+12))
The solution
You have entered
[src]
_________________
/ 2
/ - x + 7*x - 10
/ ---------------
/ 2
\/ x - 7*x + 12
$$\sqrt{\frac{- x^{2} + 7 x - 10}{x^{2} - 7 x + 12}}$$
/ _________________\ | / 2 | d | / - x + 7*x - 10 | --| / --------------- | dx| / 2 | \\/ x - 7*x + 12 /
$$\frac{d}{d x} \sqrt{\frac{- x^{2} + 7 x - 10}{x^{2} - 7 x + 12}}$$
Detail solution
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Let .
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Apply the power rule: goes to
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Then, apply the chain rule. Multiply by :
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Apply the quotient rule, which is:
and .
To find :
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Differentiate term by term:
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The derivative of the constant is zero.
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The derivative of a constant times a function is the constant times the derivative of the function.
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Apply the power rule: goes to
So, the result is:
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The derivative of a constant times a function is the constant times the derivative of the function.
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Apply the power rule: goes to
So, the result is:
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The result is:
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To find :
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Differentiate term by term:
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The derivative of the constant is zero.
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Apply the power rule: goes to
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The derivative of a constant times a function is the constant times the derivative of the function.
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Apply the power rule: goes to
So, the result is:
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The result is:
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Now plug in to the quotient rule:
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The result of the chain rule is:
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Now simplify:
The answer is:
The first derivative
[src]
_________________
/ 2 / / 2 \\
/ - x + 7*x - 10 | 7 - 2*x (7 - 2*x)*\- x + 7*x - 10/| / 2 \
/ --------------- *|----------------- + ---------------------------|*\x - 7*x + 12/
/ 2 | / 2 \ 2 |
\/ x - 7*x + 12 |2*\x - 7*x + 12/ / 2 \ |
\ 2*\x - 7*x + 12/ /
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2
- x + 7*x - 10
$$\frac{\sqrt{\frac{- x^{2} + 7 x - 10}{x^{2} - 7 x + 12}} \left(\frac{\left(- 2 x + 7\right) \left(- x^{2} + 7 x - 10\right)}{2 \left(x^{2} - 7 x + 12\right)^{2}} + \frac{- 2 x + 7}{2 \left(x^{2} - 7 x + 12\right)}\right) \left(x^{2} - 7 x + 12\right)}{- x^{2} + 7 x - 10}$$
The second derivative
[src]
/ 2 \
| / 2 \ / 2 \ / 2 \ |
| 2 | 10 + x - 7*x| 2 | 10 + x - 7*x| | 10 + x - 7*x| 2|
___________________ | (-7 + 2*x) *|-1 + -------------| (-7 + 2*x) *|-1 + -------------| |-1 + -------------| *(-7 + 2*x) |
/ / 2 \ | 2 2 2 / 2 \ | 2 | | 2 | | 2 | |
/ -\10 + x - 7*x/ | (-7 + 2*x) 10 + x - 7*x (-7 + 2*x) *\10 + x - 7*x/ \ 12 + x - 7*x/ \ 12 + x - 7*x/ \ 12 + x - 7*x/ |
/ ----------------- *|1 - ------------- - ------------- + --------------------------- + -------------------------------- - -------------------------------- + ---------------------------------|
/ 2 | 2 2 2 / 2 \ / 2 \ / 2 \ |
\/ 12 + x - 7*x | 12 + x - 7*x 12 + x - 7*x / 2 \ 2*\10 + x - 7*x/ 2*\12 + x - 7*x/ 4*\10 + x - 7*x/ |
\ \12 + x - 7*x/ /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2
10 + x - 7*x
$$\frac{\sqrt{- \frac{x^{2} - 7 x + 10}{x^{2} - 7 x + 12}} \left(\frac{\left(2 x - 7\right)^{2} \left(\frac{x^{2} - 7 x + 10}{x^{2} - 7 x + 12} - 1\right)^{2}}{4 \left(x^{2} - 7 x + 10\right)} - \frac{\left(2 x - 7\right)^{2} \left(\frac{x^{2} - 7 x + 10}{x^{2} - 7 x + 12} - 1\right)}{2 \left(x^{2} - 7 x + 12\right)} + \frac{\left(2 x - 7\right)^{2} \left(\frac{x^{2} - 7 x + 10}{x^{2} - 7 x + 12} - 1\right)}{2 \left(x^{2} - 7 x + 10\right)} + \frac{\left(2 x - 7\right)^{2} \left(x^{2} - 7 x + 10\right)}{\left(x^{2} - 7 x + 12\right)^{2}} - \frac{\left(2 x - 7\right)^{2}}{x^{2} - 7 x + 12} - \frac{x^{2} - 7 x + 10}{x^{2} - 7 x + 12} + 1\right)}{x^{2} - 7 x + 10}$$
The third derivative
[src]
/ / 2 2 2 / 2 \\ / 2 2 2 / 2 \\ / 2 / 2 \ 2 / 2 \\ 2 3 / 2 \ / 2 2 2 / 2 \\ 2 \
| 2 2 | (-7 + 2*x) 10 + x - 7*x (-7 + 2*x) *\10 + x - 7*x/| | (-7 + 2*x) 10 + x - 7*x (-7 + 2*x) *\10 + x - 7*x/| | (-7 + 2*x) 2*\10 + x - 7*x/ (-7 + 2*x) *\10 + x - 7*x/| / 2 \ / 2 \ / 2 \ | 10 + x - 7*x| | (-7 + 2*x) 10 + x - 7*x (-7 + 2*x) *\10 + x - 7*x/| / 2 \ / 2 \ |
| 10 + x - 7*x 10 + x - 7*x 2*|-1 + ------------- + ------------- - ---------------------------| 2*|-1 + ------------- + ------------- - ---------------------------| 3*|-2 + ------------- + ----------------- - ---------------------------| 2 | 10 + x - 7*x| | 10 + x - 7*x| 2 | 10 + x - 7*x| 2 3*|-1 + -------------|*|-1 + ------------- + ------------- - ---------------------------| 2 | 10 + x - 7*x| | 10 + x - 7*x| 2|
___________________ |-1 + ------------- -1 + ------------- | 2 2 2 | | 2 2 2 | | 2 2 2 | (-7 + 2*x) *|-1 + -------------| 3*|-1 + -------------| *(-7 + 2*x) |-1 + -------------| *(-7 + 2*x) | 2 | | 2 2 2 | (-7 + 2*x) *|-1 + -------------| 3*|-1 + -------------| *(-7 + 2*x) |
/ / 2 \ | 2 2 | 12 + x - 7*x 12 + x - 7*x / 2 \ | | 12 + x - 7*x 12 + x - 7*x / 2 \ | | 12 + x - 7*x 12 + x - 7*x / 2 \ | | 2 | | 2 | | 2 | \ 12 + x - 7*x/ | 12 + x - 7*x 12 + x - 7*x / 2 \ | | 2 | | 2 | |
/ -\10 + x - 7*x/ | 12 + x - 7*x 12 + x - 7*x \ \12 + x - 7*x/ / \ \12 + x - 7*x/ / \ \12 + x - 7*x/ / \ 12 + x - 7*x/ \ 12 + x - 7*x/ \ 12 + x - 7*x/ \ \12 + x - 7*x/ / \ 12 + x - 7*x/ \ 12 + x - 7*x/ |
/ ----------------- *(-7 + 2*x)*|------------------ - ------------------ - -------------------------------------------------------------------- + -------------------------------------------------------------------- + ------------------------------------------------------------------------ - -------------------------------- - ----------------------------------- - --------------------------------- + ----------------------------------------------------------------------------------------- + -------------------------------- + -----------------------------------|
/ 2 | 2 2 2 2 2 2 2 2 / 2 \ / 2 \ / 2 \ / 2 \ / 2 \ |
\/ 12 + x - 7*x | 10 + x - 7*x 12 + x - 7*x 12 + x - 7*x 10 + x - 7*x 12 + x - 7*x / 2 \ / 2 \ / 2 \ 2*\10 + x - 7*x/ \10 + x - 7*x/*\12 + x - 7*x/ 4*\10 + x - 7*x/*\12 + x - 7*x/ |
\ \10 + x - 7*x/ 4*\10 + x - 7*x/ 8*\10 + x - 7*x/ /
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2
10 + x - 7*x
$$\frac{\sqrt{- \frac{x^{2} - 7 x + 10}{x^{2} - 7 x + 12}} \cdot \left(2 x - 7\right) \left(- \frac{\left(2 x - 7\right)^{2} \left(\frac{x^{2} - 7 x + 10}{x^{2} - 7 x + 12} - 1\right)^{3}}{8 \left(x^{2} - 7 x + 10\right)^{2}} + \frac{3 \left(2 x - 7\right)^{2} \left(\frac{x^{2} - 7 x + 10}{x^{2} - 7 x + 12} - 1\right)^{2}}{4 \left(x^{2} - 7 x + 10\right) \left(x^{2} - 7 x + 12\right)} - \frac{3 \left(2 x - 7\right)^{2} \left(\frac{x^{2} - 7 x + 10}{x^{2} - 7 x + 12} - 1\right)^{2}}{4 \left(x^{2} - 7 x + 10\right)^{2}} + \frac{\left(2 x - 7\right)^{2} \left(\frac{x^{2} - 7 x + 10}{x^{2} - 7 x + 12} - 1\right)}{\left(x^{2} - 7 x + 10\right) \left(x^{2} - 7 x + 12\right)} - \frac{\left(2 x - 7\right)^{2} \left(\frac{x^{2} - 7 x + 10}{x^{2} - 7 x + 12} - 1\right)}{\left(x^{2} - 7 x + 10\right)^{2}} + \frac{3 \left(\frac{x^{2} - 7 x + 10}{x^{2} - 7 x + 12} - 1\right) \left(- \frac{\left(2 x - 7\right)^{2} \left(x^{2} - 7 x + 10\right)}{\left(x^{2} - 7 x + 12\right)^{2}} + \frac{\left(2 x - 7\right)^{2}}{x^{2} - 7 x + 12} + \frac{x^{2} - 7 x + 10}{x^{2} - 7 x + 12} - 1\right)}{2 \left(x^{2} - 7 x + 10\right)} - \frac{\frac{x^{2} - 7 x + 10}{x^{2} - 7 x + 12} - 1}{x^{2} - 7 x + 12} + \frac{\frac{x^{2} - 7 x + 10}{x^{2} - 7 x + 12} - 1}{x^{2} - 7 x + 10} - \frac{2 \left(- \frac{\left(2 x - 7\right)^{2} \left(x^{2} - 7 x + 10\right)}{\left(x^{2} - 7 x + 12\right)^{2}} + \frac{\left(2 x - 7\right)^{2}}{x^{2} - 7 x + 12} + \frac{x^{2} - 7 x + 10}{x^{2} - 7 x + 12} - 1\right)}{x^{2} - 7 x + 12} + \frac{3 \left(- \frac{\left(2 x - 7\right)^{2} \left(x^{2} - 7 x + 10\right)}{\left(x^{2} - 7 x + 12\right)^{2}} + \frac{\left(2 x - 7\right)^{2}}{x^{2} - 7 x + 12} + \frac{2 \left(x^{2} - 7 x + 10\right)}{x^{2} - 7 x + 12} - 2\right)}{x^{2} - 7 x + 12} + \frac{2 \left(- \frac{\left(2 x - 7\right)^{2} \left(x^{2} - 7 x + 10\right)}{\left(x^{2} - 7 x + 12\right)^{2}} + \frac{\left(2 x - 7\right)^{2}}{x^{2} - 7 x + 12} + \frac{x^{2} - 7 x + 10}{x^{2} - 7 x + 12} - 1\right)}{x^{2} - 7 x + 10}\right)}{x^{2} - 7 x + 10}$$
The graph