2. Find the derivatives of the following functions. $$ \begin{array}{ll} ext { a) } y=\left(5-2 x^{2} ight)^{6} & ext { b) } y=\frac{1}{(\sqrt{x}-x)^{3}} \ ext { c) } y=\frac{3}{2 \sqrt{x+1}} & ext { d) } y=(4 x-3)(1-2 x)^{4} \ ext { e) } y=\frac{2 x-3}{x^{2}-5} & ext { f) } y=\left(x^{3}+1 ight) \sqrt{2 x+7}\end{array} $$

Question

2. Find the derivatives of the following functions. $$ \begin{array}{ll}	ext { a) } y=\left(5-2 x^{2}ight)^{6} & 	ext { b) } y=\frac{1}{(\sqrt{x}-x)^{3}} \ 	ext { c) } y=\frac{3}{2 \sqrt{x+1}} & 	ext { d) } y=(4 x-3)(1-2 x)^{4} \ 	ext { e) } y=\frac{2 x-3}{x^{2}-5} & 	ext { f) } y=\left(x^{3}+1ight) \sqrt{2 x+7}\end{array} $$

UpStudy AI · Accepted Answer

Let's find the derivatives of each function step by step.

### **a) $$ y = (5 - 2x^{2})^{6} $$**

To differentiate $$ y = (5 - 2x^{2})^{6} $$, we'll use the **chain rule**.

**Chain Rule:** If $$ y = [u(x)]^n $$, then $$ y' = n \cdot [u(x)]^{n-1} \cdot u'(x) $$.

1. **Identify the outer function and the inner function:**
   - Outer function: $$ [u]^6 $$
   - Inner function: $$ u = 5 - 2x^{2} $$

2. **Differentiate the outer function with respect to the inner function:**
   $$
   \frac{d}{du} [u^{6}] = 6u^{5}
   $$

3. **Differentiate the inner function with respect to $$ x $$:**
   $$
   \frac{du}{dx} = \frac{d}{dx} (5 - 2x^{2}) = -4x
   $$

4. **Apply the chain rule:**
   $$
   \frac{dy}{dx} = 6u^{5} \cdot (-4x) = 6(5 - 2x^{2})^{5} \cdot (-4x)
   $$

5. **Simplify the expression:**
   $$
   \frac{dy}{dx} = -24x (5 - 2x^{2})^{5}
   $$

**Final Answer:**
$$
\boxed{ \frac{dy}{dx} = -24x\, (5 - 2 x^{2})^{5} }
$$
The derivative of $$ y = (5 - 2x^{2})^{6} $$ is $$ \frac{dy}{dx} = -24x\, (5 - 2 x^{2})^{5} $$.

Find the derivatives of the following functions.

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