(\partial)/(\partial x)(4x^{-x^2-y^2-z^2})

⚠ (\partial)/(\partial x)(4x^{-x^2-y^2-z^2}) is read as dif(4x^(-x^2-y^2-z^2),x): use dif(expression, variable) for a derivative with respect to that variable, use (numerator)/(denominator) for a quotient, use ^(...) to group an exponent, use * to multiply

1) Differentiate with respect to x while holding y, z constant

Let's perform the differentiation:

\frac{\partial}{\partial x} (4 x^{- x^{2} - y^{2} - z^{2}})

To differentiate on intervals where

x > 0

, write

x^{- x^{2} - y^{2} - z^{2}}

e^{(- x^{2} - y^{2} - z^{2}) \cdot \ln (x)}

ⅰ

\frac{\partial}{\partial x} (x^{- x^{2} - y^{2} - z^{2}})

\frac{\partial}{\partial x} ((- x^{2} - y^{2} - z^{2}) \cdot \ln (x)) \cdot e^{(- x^{2} - y^{2} - z^{2}) \cdot \ln (x)}

4 \cdot \frac{\partial}{\partial x} ((- x^{2} - y^{2} - z^{2}) \cdot \ln (x)) \cdot e^{(- x^{2} - y^{2} - z^{2}) \cdot \ln (x)}

For

u1

(- x^{2} - y^{2} - z^{2})

and

v2

\ln (x)

, use the differentiation rule:

\frac{\partial}{\partial x} (u1 \cdot v2) = \frac{\partial}{\partial x} (u1) \cdot v2 + u1 \cdot \frac{\partial}{\partial x} (v2)

2) Let's perform the differentiation

u1

\frac{\partial}{\partial x} (- x^{2} - y^{2} - z^{2})

ⓘ The derivative of a sum is the sum of the derivatives:

\frac{\partial}{\partial x} (- x^{2} - y^{2} - z^{2})

\frac{\partial}{\partial x} (- x^{2}) + \frac{\partial}{\partial x} (- y^{2}) + \frac{\partial}{\partial x} (- z^{2})

\frac{\partial}{\partial x} (- x^{2}) + \frac{\partial}{\partial x} (- y^{2}) + \frac{\partial}{\partial x} (- z^{2})

ⓘ The derivative of x^n is n*x^(n-1):

\frac{\partial}{\partial x} (- x^{2})

- 2 x

■The derivative of a constant is zero:

\frac{\partial}{\partial x} (- y^{2})

0

\frac{\partial}{\partial x} (- z^{2})

0

- 2 x + 0 + 0

ⓘ Adding 0 changes nothing:

- 2 x + 0 + 0

- 2 x

- 2 x

3) Let's perform the differentiation

v2

\frac{\partial}{\partial x} (\ln (x))

ⓘ The derivative of ln(x) is 1/x:

\frac{\partial}{\partial x} (\ln (x))

\frac{1}{x}

\frac{1}{x}

4) Let's evaluate the resulting derivative (

u1 \cdot v2

)':

- 2 x \cdot \ln (x) + (- x^{2} - y^{2} - z^{2}) \cdot \frac{1}{x}

ⓘ Multiplying a fraction by a term multiplies its numerator:

(- x^{2} - y^{2} - z^{2}) \cdot \frac{1}{x}

\frac{(- x^{2} - y^{2} - z^{2}) \cdot 1}{x}

\frac{- x^{2} - y^{2} - z^{2}}{x}

■The minus sign is factored out, changing the sign of each term:

- x^{2} - y^{2} - z^{2}

- (x^{2} + y^{2} + z^{2})

- 2 x \cdot \ln (x) - \frac{x^{2} + y^{2} + z^{2}}{x}

5) You can then resume the differentiation:

ⅰ

\frac{\partial}{\partial x} ((- x^{2} - y^{2} - z^{2}) \cdot \ln (x))

- 2 x \cdot \ln (x) - \frac{x^{2} + y^{2} + z^{2}}{x}

4 \cdot (- 2 x \cdot \ln (x) - \frac{x^{2} + y^{2} + z^{2}}{x}) \cdot x^{- x^{2} - y^{2} - z^{2}}

ⓘ The factor is moved to the numerator or the denominator by changing the sign of its exponent, for a nonzero base:

4 \cdot (- 2 x \cdot \ln (x) - \frac{x^{2} + y^{2} + z^{2}}{x}) \cdot x^{- x^{2} - y^{2} - z^{2}}

\frac{4 \cdot (- 2 x \cdot \ln (x) - \frac{x^{2} + y^{2} + z^{2}}{x})}{x^{x^{2} + y^{2} + z^{2}}}

\frac{4 \cdot (- 2 x \cdot \ln (x) - \frac{x^{2} + y^{2} + z^{2}}{x})}{x^{x^{2} + y^{2} + z^{2}}}

ⓘ The minus sign is factored out, changing the sign of each term:

- 2 x \cdot \ln (x) - \frac{x^{2} + y^{2} + z^{2}}{x}

- (2 x \cdot \ln (x) + \frac{x^{2} + y^{2} + z^{2}}{x})

\frac{4 \cdot (- (2 x \cdot \ln (x) + \frac{x^{2} + y^{2} + z^{2}}{x}))}{x^{x^{2} + y^{2} + z^{2}}}

ⓘ The minus sign is placed in front of the product:

4 \cdot (- (2 x \cdot \ln (x) + \frac{x^{2} + y^{2} + z^{2}}{x}))

- 4 \cdot (2 x \cdot \ln (x) + \frac{x^{2} + y^{2} + z^{2}}{x})

- \frac{4 \cdot (2 x \cdot \ln (x) + \frac{x^{2} + y^{2} + z^{2}}{x})}{x^{x^{2} + y^{2} + z^{2}}}

ⓘ To add fractions, put them over the same denominator, then add the numerators:

2 x \cdot \ln (x) + \frac{x^{2} + y^{2} + z^{2}}{x}

\frac{2 x \cdot \ln (x) \cdot x + (x^{2} + y^{2} + z^{2})}{x}

- \frac{4 \cdot (\frac{2 x \cdot \ln (x) \cdot x + (x^{2} + y^{2} + z^{2})}{x})}{x^{x^{2} + y^{2} + z^{2}}}

ⓘ Multiplying a fraction by a number multiplies its numerator:

4 \cdot \frac{2 \cdot x \cdot \ln (x) \cdot x + x^{2} + y^{2} + z^{2}}{x}

\frac{4 \cdot (2 \cdot x \cdot \ln (x) \cdot x + x^{2} + y^{2} + z^{2})}{x}

■Multiplying powers with the same base adds the exponents:

x \cdot x

x^{2}

- \frac{\frac{4 \cdot (2 x^{2} \cdot \ln (x) + x^{2} + y^{2} + z^{2})}{x}}{x^{x^{2} + y^{2} + z^{2}}}

ⓘ Dividing a fraction by

x^{x^{2} + y^{2} + z^{2}}

is multiplying its denominator by

x^{x^{2} + y^{2} + z^{2}}

- \frac{\frac{4 \cdot (2 x^{2} \cdot \ln (x) + x^{2} + y^{2} + z^{2})}{x}}{x^{x^{2} + y^{2} + z^{2}}}

- \frac{4 \cdot (2 x^{2} \cdot \ln (x) + x^{2} + y^{2} + z^{2})}{x x^{x^{2} + y^{2} + z^{2}}}

■Multiplying powers with the same base adds the exponents:

x x^{x^{2} + y^{2} + z^{2}}

x^{x^{2} + y^{2} + z^{2} + 1}

▶Derivative:

- \frac{4 \cdot (2 x^{2} \cdot \ln (x) + x^{2} + y^{2} + z^{2})}{x^{x^{2} + y^{2} + z^{2} + 1}}