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How to use it?
- Derivative of:
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Derivative of e^(-x)
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Derivative of tanh(x)
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Derivative of (x^2+1)^3
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Derivative of cos(x)^(3)
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Identical expressions
- (x^ two - four)/(two *sqrt(four -(x^ two)))*arctg(sqrt(two *x))-(one / two)*(ln(four -(x^ two)))^ two
- (x squared minus 4) divide by (2 multiply by square root of (4 minus (x squared ))) multiply by arctg( square root of (2 multiply by x)) minus (1 divide by 2) multiply by (ln(4 minus (x squared ))) squared
- (x to the power of two minus four) divide by (two multiply by square root of (four minus (x to the power of two))) multiply by arctg( square root of (two multiply by x)) minus (one divide by two) multiply by (ln(four minus (x to the power of two))) to the power of two
- (x^2-4)/(2*√(4-(x^2)))*arctg(√(2*x))-(1/2)*(ln(4-(x^2)))^2
- (x2-4)/(2*sqrt(4-(x2)))*arctg(sqrt(2*x))-(1/2)*(ln(4-(x2)))2
- x2-4/2*sqrt4-x2*arctgsqrt2*x-1/2*ln4-x22
- (x²-4)/(2*sqrt(4-(x²)))*arctg(sqrt(2*x))-(1/2)*(ln(4-(x²)))²
- (x to the power of 2-4)/(2*sqrt(4-(x to the power of 2)))*arctg(sqrt(2*x))-(1/2)*(ln(4-(x to the power of 2))) to the power of 2
- (x^2-4)/(2sqrt(4-(x^2)))arctg(sqrt(2x))-(1/2)(ln(4-(x^2)))^2
- (x2-4)/(2sqrt(4-(x2)))arctg(sqrt(2x))-(1/2)(ln(4-(x2)))2
- x2-4/2sqrt4-x2arctgsqrt2x-1/2ln4-x22
- x^2-4/2sqrt4-x^2arctgsqrt2x-1/2ln4-x^2^2
- (x^2-4) divide by (2*sqrt(4-(x^2)))*arctg(sqrt(2*x))-(1 divide by 2)*(ln(4-(x^2)))^2
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Similar expressions
- (x^2+4)/(2*sqrt(4-(x^2)))*arctg(sqrt(2*x))-(1/2)*(ln(4-(x^2)))^2
- (x^2-4)/(2*sqrt(4+(x^2)))*arctg(sqrt(2*x))-(1/2)*(ln(4-(x^2)))^2
- (x^2-4)/(2*sqrt(4-(x^2)))*arctg(sqrt(2*x))+(1/2)*(ln(4-(x^2)))^2
- (x^2-4)/(2*sqrt(4-(x^2)))*arctg(sqrt(2*x))-(1/2)*(ln(4+(x^2)))^2
Derivative of (x^2-4)/(2*sqrt(4-(x^2)))*arctg(sqrt(2*x))-(1/2)*(ln(4-(x^2)))^2
The solution
You have entered
[src]
2 2/ 2\
x - 4 / _____\ log \4 - x /
-------------*atan\\/ 2*x / - ------------
________ 2
/ 2
2*\/ 4 - x
$$\frac{x^{2} - 4}{2 \sqrt{4 - x^{2}}} \operatorname{atan}{\left(\sqrt{2 x} \right)} - \frac{\log{\left(4 - x^{2} \right)}^{2}}{2}$$
((x^2 - 4)/((2*sqrt(4 - x^2))))*atan(sqrt(2*x)) - log(4 - x^2)^2/2
The first derivative
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___ 1 / 2 \
\/ 2 *-------------*\x - 4/
________
/ / 2 \ \ / 2\ / 2
| 1 x*\x - 4/ | / _____\ 2*x*log\4 - x / 2*\/ 4 - x
|2*x*------------- + -------------|*atan\\/ 2*x / + --------------- + ----------------------------
| ________ 3/2| 2 ___
| / 2 / 2\ | 4 - x 2*\/ x *(1 + 2*x)
\ 2*\/ 4 - x 2*\4 - x / /
$$\frac{2 x \log{\left(4 - x^{2} \right)}}{4 - x^{2}} + \left(2 x \frac{1}{2 \sqrt{4 - x^{2}}} + \frac{x \left(x^{2} - 4\right)}{2 \left(4 - x^{2}\right)^{\frac{3}{2}}}\right) \operatorname{atan}{\left(\sqrt{2 x} \right)} + \frac{\sqrt{2} \frac{1}{2 \sqrt{4 - x^{2}}} \left(x^{2} - 4\right)}{2 \sqrt{x} \left(2 x + 1\right)}$$
The second derivative
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/ 2 2 2 / 2\\
| -4 + x 4*x 3*x *\-4 + x /| / _____\ / 2\
|2 + ------- + ------ + --------------|*atan\\/ 2*x / ___ ___ | -4 + x |
| 2 2 2 | \/ 2 *\/ x *|2 + -------|
2 / 2\ | 4 - x 4 - x / 2\ | 2 / 2\ ___ ___ ___ / 2\ ___ / 2\ ___ ___ / 2\ | 2|
4*x 2*log\4 - x / \ \4 - x / / 4*x *log\4 - x / \/ 2 *\/ x \/ 2 *\-4 + x / \/ 2 *\-4 + x / \/ 2 *\/ x *\-4 + x / \ 4 - x /
- ---------- - ------------- + ----------------------------------------------------- + ---------------- + ----------------------- - ------------------------------ - ---------------------------- + ----------------------- + -------------------------
2 2 ________ 2 ________ ________ ________ 3/2 ________
/ 2\ -4 + x / 2 / 2\ / 2 ___ 2 / 2 3/2 / 2 / 2\ / 2
\-4 + x / 2*\/ 4 - x \-4 + x / 2*(1 + 2*x)*\/ 4 - x 2*\/ x *(1 + 2*x) *\/ 4 - x 8*x *(1 + 2*x)*\/ 4 - x 4*(1 + 2*x)*\4 - x / 4*(1 + 2*x)*\/ 4 - x
$$\frac{\sqrt{2} \sqrt{x} \left(2 + \frac{x^{2} - 4}{4 - x^{2}}\right)}{4 \sqrt{4 - x^{2}} \left(2 x + 1\right)} + \frac{\sqrt{2} \sqrt{x}}{2 \sqrt{4 - x^{2}} \left(2 x + 1\right)} + \frac{\sqrt{2} \sqrt{x} \left(x^{2} - 4\right)}{4 \left(4 - x^{2}\right)^{\frac{3}{2}} \left(2 x + 1\right)} + \frac{4 x^{2} \log{\left(4 - x^{2} \right)}}{\left(x^{2} - 4\right)^{2}} - \frac{4 x^{2}}{\left(x^{2} - 4\right)^{2}} - \frac{2 \log{\left(4 - x^{2} \right)}}{x^{2} - 4} + \frac{\left(\frac{4 x^{2}}{4 - x^{2}} + \frac{3 x^{2} \left(x^{2} - 4\right)}{\left(4 - x^{2}\right)^{2}} + 2 + \frac{x^{2} - 4}{4 - x^{2}}\right) \operatorname{atan}{\left(\sqrt{2 x} \right)}}{2 \sqrt{4 - x^{2}}} - \frac{\sqrt{2} \left(x^{2} - 4\right)}{2 \sqrt{x} \sqrt{4 - x^{2}} \left(2 x + 1\right)^{2}} - \frac{\sqrt{2} \left(x^{2} - 4\right)}{8 x^{\frac{3}{2}} \sqrt{4 - x^{2}} \left(2 x + 1\right)}$$
The third derivative
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/ / 2\ 2 2 / 2\\ / 2 2 2 / 2\\
| 3*\-4 + x / 6*x 5*x *\-4 + x /| / _____\ ___ | -4 + x 4*x 3*x *\-4 + x /| / 2\ / 2\
3*x*|4 + ----------- + ------ + --------------|*atan\\/ 2*x / \/ 2 *|2 + ------- + ------ + --------------| ___ ___ | -4 + x | ___ | -4 + x |
| 2 2 2 | | 2 2 2 | \/ 2 *\/ x *|2 + -------| \/ 2 *|2 + -------|
3 3 / 2\ / 2\ ___ 3/2 ___ ___ | 4 - x 4 - x / 2\ | ___ / 2\ | 4 - x 4 - x / 2\ | ___ ___ / 2\ ___ / 2\ | 2| | 2| ___ 3/2 / 2\ ___ / 2\
12*x 24*x 16*x *log\4 - x / 12*x*log\4 - x / \/ 2 *x 2*\/ 2 *\/ x \ \4 - x / / \/ 2 *\-4 + x / \ \4 - x / / \/ 2 *\/ x *\-4 + x / 2*\/ 2 *\-4 + x / \ 4 - x / \ 4 - x / 3*\/ 2 *x *\-4 + x / 3*\/ 2 *\-4 + x /
- ---------- + ---------- - ----------------- + ---------------- + --------------------- - ---------------------- + ------------------------------------------------------------- + ----------------------------- + --------------------------------------------- - ---------------------- + ---------------------------- - ------------------------- - ----------------------------- + ----------------------- + -----------------------------
2 3 3 2 3/2 ________ 3/2 ________ ________ 3/2 ________ ________ ________ 5/2 ________
/ 2\ / 2\ / 2\ / 2\ / 2\ 2 / 2 / 2\ 3/2 2 / 2 ___ / 2 2 / 2\ ___ 3 / 2 2 / 2 ___ / 2 / 2\ 5/2 / 2
\-4 + x / \-4 + x / \-4 + x / \-4 + x / (1 + 2*x)*\4 - x / (1 + 2*x) *\/ 4 - x 2*\4 - x / 2*x *(1 + 2*x) *\/ 4 - x 2*\/ x *(1 + 2*x)*\/ 4 - x (1 + 2*x) *\4 - x / \/ x *(1 + 2*x) *\/ 4 - x 2*(1 + 2*x) *\/ 4 - x 8*\/ x *(1 + 2*x)*\/ 4 - x 4*(1 + 2*x)*\4 - x / 16*x *(1 + 2*x)*\/ 4 - x
$$\frac{\sqrt{2} x^{\frac{3}{2}}}{\left(4 - x^{2}\right)^{\frac{3}{2}} \left(2 x + 1\right)} + \frac{3 \sqrt{2} x^{\frac{3}{2}} \left(x^{2} - 4\right)}{4 \left(4 - x^{2}\right)^{\frac{5}{2}} \left(2 x + 1\right)} - \frac{\sqrt{2} \sqrt{x} \left(2 + \frac{x^{2} - 4}{4 - x^{2}}\right)}{2 \sqrt{4 - x^{2}} \left(2 x + 1\right)^{2}} - \frac{2 \sqrt{2} \sqrt{x}}{\sqrt{4 - x^{2}} \left(2 x + 1\right)^{2}} - \frac{\sqrt{2} \sqrt{x} \left(x^{2} - 4\right)}{\left(4 - x^{2}\right)^{\frac{3}{2}} \left(2 x + 1\right)^{2}} - \frac{16 x^{3} \log{\left(4 - x^{2} \right)}}{\left(x^{2} - 4\right)^{3}} + \frac{24 x^{3}}{\left(x^{2} - 4\right)^{3}} + \frac{12 x \log{\left(4 - x^{2} \right)}}{\left(x^{2} - 4\right)^{2}} - \frac{12 x}{\left(x^{2} - 4\right)^{2}} + \frac{3 x \left(\frac{6 x^{2}}{4 - x^{2}} + \frac{5 x^{2} \left(x^{2} - 4\right)}{\left(4 - x^{2}\right)^{2}} + 4 + \frac{3 \left(x^{2} - 4\right)}{4 - x^{2}}\right) \operatorname{atan}{\left(\sqrt{2 x} \right)}}{2 \left(4 - x^{2}\right)^{\frac{3}{2}}} - \frac{\sqrt{2} \left(2 + \frac{x^{2} - 4}{4 - x^{2}}\right)}{8 \sqrt{x} \sqrt{4 - x^{2}} \left(2 x + 1\right)} + \frac{\sqrt{2} \left(\frac{4 x^{2}}{4 - x^{2}} + \frac{3 x^{2} \left(x^{2} - 4\right)}{\left(4 - x^{2}\right)^{2}} + 2 + \frac{x^{2} - 4}{4 - x^{2}}\right)}{2 \sqrt{x} \sqrt{4 - x^{2}} \left(2 x + 1\right)} + \frac{2 \sqrt{2} \left(x^{2} - 4\right)}{\sqrt{x} \sqrt{4 - x^{2}} \left(2 x + 1\right)^{3}} + \frac{\sqrt{2} \left(x^{2} - 4\right)}{2 x^{\frac{3}{2}} \sqrt{4 - x^{2}} \left(2 x + 1\right)^{2}} + \frac{3 \sqrt{2} \left(x^{2} - 4\right)}{16 x^{\frac{5}{2}} \sqrt{4 - x^{2}} \left(2 x + 1\right)}$$