Question1) $y=\left(x^{3}+3\right)^{5}$

Studdy Solution

STEP 1

1. We are given the function $y = (x^3 + 3)^5$ .
2. We need to find the derivative of $y$ with respect to $x$ .

STEP 2

1. Identify the outer and inner functions for the chain rule.
2. Apply the chain rule to differentiate the function.
3. Simplify the derivative expression.

STEP 3

Identify the outer and inner functions for the chain rule:
The outer function is $u^5$ where $u = x^3 + 3$ .
The inner function is $u = x^3 + 3$ .

STEP 4

Apply the chain rule to differentiate the function:
The chain rule states that if $y = f(g(x))$ , then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$ .
For our function, $y = u^5$ and $u = x^3 + 3$ .
Differentiate the outer function with respect to $u$ :
$\frac{d}{du}(u^5) = 5u^4$
Differentiate the inner function with respect to $x$ :
$\frac{d}{dx}(x^3 + 3) = 3x^2$

STEP 5

Combine the derivatives using the chain rule:
$\frac{dy}{dx} = 5u^4 \cdot 3x^2$
Substitute back $u = x^3 + 3$ :
$\frac{dy}{dx} = 5(x^3 + 3)^4 \cdot 3x^2$
Simplify the expression:
$\frac{dy}{dx} = 15x^2(x^3 + 3)^4$
The derivative of $y$ with respect to $x$ is:
$\boxed{15x^2(x^3 + 3)^4}$

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