Mister Exam
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- Derivative of a implicit defined function
- Derivative of Parametric Function
- Partial derivative of the function
- Curve tracing functions Step by Step
- Integral Step by Step
- Differential equations Step by Step
- Limits Step by Step
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How to use it?
- Derivative of:
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Derivative of x^2+4x
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Derivative of sin(x)^(9)
- Derivative of i*n*sin(x)
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Derivative of sin(x)-x*cos(x)
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Identical expressions
- sin(x)^(one / two)
- sinus of (x) to the power of (1 divide by 2)
- sinus of (x) to the power of (one divide by two)
- sin(x)(1/2)
- sinx1/2
- sinx^1/2
- sin(x)^(1 divide by 2)
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Similar expressions
- sinx^1/2
- y=(x^3+1)^sin(x^1/2)
- (sin(x))^(1/2)
- sinx^(1/2)
Derivative of sin(x)^(1/2)
The solution
You have entered
[src]
________ \/ sin(x)
$$\sqrt{\sin{\left(x \right)}}$$
d / ________\ --\\/ sin(x) / dx
$$\frac{d}{d x} \sqrt{\sin{\left(x \right)}}$$
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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The derivative of sine is cosine:
The result of the chain rule is:
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The answer is:
The first derivative
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cos(x)
------------
________
2*\/ sin(x)
$$\frac{\cos{\left(x \right)}}{2 \sqrt{\sin{\left(x \right)}}}$$
The second derivative
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/ 2 \
| ________ cos (x) |
-|2*\/ sin(x) + ---------|
| 3/2 |
\ sin (x)/
----------------------------
4
$$- \frac{2 \sqrt{\sin{\left(x \right)}} + \frac{\cos^{2}{\left(x \right)}}{\sin^{\frac{3}{2}}{\left(x \right)}}}{4}$$
The third derivative
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/ 2 \
| 3*cos (x)|
|2 + ---------|*cos(x)
| 2 |
\ sin (x) /
----------------------
________
8*\/ sin(x)
$$\frac{\left(2 + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{8 \sqrt{\sin{\left(x \right)}}}$$
The graph