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Exercise

$\frac{d}{dx}\:ln\left(3e^xy\:-\:z\right)$

Step-by-step Solution

1

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{3e^xy-z}\frac{d}{dx}\left(3e^xy-z\right)$
2

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{1}{3e^xy-z}\frac{d}{dx}\left(3e^xy\right)$
3

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$3\left(\frac{1}{3e^xy-z}\right)y\frac{d}{dx}\left(e^x\right)$
4

Applying the derivative of the exponential function

$3\left(\frac{1}{3e^xy-z}\right)ye^x$
5

Multiply the fraction by the term

$\frac{3ye^x}{3e^xy-z}$

Final answer to the exercise

$\frac{3ye^x}{3e^xy-z}$

Try other ways to solve this exercise

  • Choose an option
  • Find the derivative using the definition
  • Find the derivative using the product rule
  • Find the derivative using the quotient rule
  • Find the derivative using logarithmic differentiation
  • Find the derivative
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
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