Question

question: find the derivative of y=(6 - 5x^3)^x. be sure to include parentheses around the arguments of any logarithmic functions. provide your answer below. y=

Explanation:

Step1: Apply the product - rule

The product - rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Let $u = x$ and $v=(6 - 5x^{3})$. First, find the derivative of $u$ and $v$. The derivative of $u=x$ with respect to $x$ is $u' = 1$.

Step2: Apply the chain - rule to find $v'$

Let $t = 6-5x^{3}$, so $v = t$. The derivative of $v$ with respect to $t$ is $\frac{dv}{dt}=1$, and the derivative of $t$ with respect to $x$ is $\frac{dt}{dx}=-15x^{2}$. By the chain - rule $\frac{dv}{dx}=\frac{dv}{dt}\cdot\frac{dt}{dx}=-15x^{2}$.

Step3: Calculate $y'$

Using the product - rule $y'=u'v + uv'$, substitute $u = x$, $u' = 1$, $v=(6 - 5x^{3})$, and $v'=-15x^{2}$ into the formula.
$y'=1\cdot(6 - 5x^{3})+x\cdot(-15x^{2})$.

Step4: Simplify the expression

$y'=6 - 5x^{3}-15x^{3}=6 - 20x^{3}$.

Answer:

$6 - 20x^{3}$