Question

section 2.6: chain rule (homework)
score: 110/170 answered: 11/17
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question 12
0/10 pts 4 99 details
use the chain rule to find the derivative of 6\sqrt{3x^{4}+9x^{5}}
type your answer without fractional or negative exponents. use sqrt(x) for \sqrt{x}.

Explanation:

Step1: Rewrite the function

Let $y = 6\sqrt{3x^{4}+9x^{5}}=6(3x^{4}+9x^{5})^{\frac{1}{2}}$.

Step2: Apply the chain - rule

The chain - rule states that if $y = f(g(x))$, then $y^\prime=f^\prime(g(x))\cdot g^\prime(x)$. Let $u = 3x^{4}+9x^{5}$, so $y = 6u^{\frac{1}{2}}$. First, find $\frac{dy}{du}$ and $\frac{du}{dx}$.
For $y = 6u^{\frac{1}{2}}$, $\frac{dy}{du}=6\times\frac{1}{2}u^{-\frac{1}{2}} = 3u^{-\frac{1}{2}}$.
For $u = 3x^{4}+9x^{5}$, $\frac{du}{dx}=3\times4x^{3}+9\times5x^{4}=12x^{3}+45x^{4}$.

Step3: Calculate $\frac{dy}{dx}$

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u = 3x^{4}+9x^{5}$ back into $\frac{dy}{du}$:
$\frac{dy}{dx}=3(3x^{4}+9x^{5})^{-\frac{1}{2}}\cdot(12x^{3}+45x^{4})$.

Step4: Simplify the expression

$\frac{dy}{dx}=\frac{3(12x^{3}+45x^{4})}{\sqrt{3x^{4}+9x^{5}}}=\frac{36x^{3}+135x^{4}}{\sqrt{3x^{4}+9x^{5}}}$.

Answer:

$\frac{36x^{3}+135x^{4}}{\sqrt{3x^{4}+9x^{5}}}$