Math  /  Calculus

QuestionFind the derivative of the function f(x)=(5x5)2(4x2+1)3 f(x) = (5x - 5)^2(4x^2 + 1)^3 .

Studdy Solution

STEP 1

What is this asking? We need to find the *derivative* of a function that's a *product* of two expressions, each raised to a power. Watch out! Don't forget the chain rule when dealing with those powers, and remember the product rule, too!

STEP 2

1. Define the function
2. Apply the product rule
3. Calculate the derivatives of the individual factors
4. Substitute and simplify

STEP 3

Alright, let's **define** our function!
We have f(x)=(5x5)2(4x2+1)3f(x) = (5x - 5)^2 (4x^2 + 1)^3.
This is a *product* of two functions, so we'll need the *product rule*!

STEP 4

Remember, the **product rule** says if we have two functions, let's call them uu and vv, then the derivative of their product is (uv)=uv+uv(u \cdot v)' = u' \cdot v + u \cdot v'.

STEP 5

In our case, u=(5x5)2u = (5x - 5)^2 and v=(4x2+1)3v = (4x^2 + 1)^3.
So, we need to find uu' and vv'!

STEP 6

Let's find uu'.
We have u=(5x5)2u = (5x - 5)^2.
Using the **chain rule**, we get u=2(5x5)1(5)=10(5x5)=50x50u' = 2(5x - 5)^1 \cdot (5) = 10(5x - 5) = 50x - 50.
See how the **exponent** came down, and we multiplied by the **derivative of the inside**?

STEP 7

Now for vv'.
We have v=(4x2+1)3v = (4x^2 + 1)^3.
Again, using the **chain rule**, we get v=3(4x2+1)2(8x)=24x(4x2+1)2v' = 3(4x^2 + 1)^2 \cdot (8x) = 24x(4x^2 + 1)^2.
We brought the **exponent** down and multiplied by the **derivative of the inside**, just like before!

STEP 8

Now, let's **plug everything** back into the product rule: f(x)=uv+uvf'(x) = u' \cdot v + u \cdot v' f(x)=(50x50)(4x2+1)3+(5x5)2(24x(4x2+1)2)f'(x) = (50x - 50)(4x^2 + 1)^3 + (5x - 5)^2(24x(4x^2 + 1)^2)

STEP 9

Let's **factor out** common terms.
Notice both terms have (5x5)(5x - 5) and (4x2+1)2(4x^2 + 1)^2.
We can rewrite (50x50)(50x - 50) as 10(5x5)10(5x - 5). f(x)=10(5x5)(4x2+1)3+(5x5)224x(4x2+1)2f'(x) = 10(5x - 5)(4x^2 + 1)^3 + (5x - 5)^2 \cdot 24x(4x^2 + 1)^2 f(x)=(5x5)(4x2+1)2[10(4x2+1)+24x(5x5)]f'(x) = (5x - 5)(4x^2 + 1)^2 [10(4x^2 + 1) + 24x(5x - 5)]f(x)=(5x5)(4x2+1)2(40x2+10+120x2120x)f'(x) = (5x - 5)(4x^2 + 1)^2 (40x^2 + 10 + 120x^2 - 120x)f(x)=(5x5)(4x2+1)2(160x2120x+10)f'(x) = (5x - 5)(4x^2 + 1)^2 (160x^2 - 120x + 10)f(x)=10(5x5)(4x2+1)2(16x212x+1)f'(x) = 10(5x - 5)(4x^2 + 1)^2 (16x^2 - 12x + 1)

STEP 10

The derivative of f(x)f(x) is 10(5x5)(4x2+1)2(16x212x+1)10(5x - 5)(4x^2 + 1)^2 (16x^2 - 12x + 1).

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