Question

Найти f'(x)1)f(x) =1/(5x+1)^32)f(x)=(9x+5)^4

Answer 1

f(g(x))' = f'(g(x))·g'(x)

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Answer 2

To find the derivatives of the given functions, we can use the power rule and the chain rule. Let's find the derivatives step by step:

1. f(x) = 1/(5x + 1)^3

Step 1: Apply the power rule to the expression (5x + 1)^3: d/dx [(5x + 1)^3] = 3(5x + 1)^(3-1) * d/dx (5x + 1) = 3(5x + 1)^2 * 5 = 15(5x + 1)^2

Step 2: Now, we have to apply the chain rule to the whole expression f(x) = 1/(5x + 1)^3: f'(x) = d/dx [1/(5x + 1)^3] = -1/(5x + 1)^4 * d/dx (5x + 1) = -1/(5x + 1)^4 * 5 = -5/(5x + 1)^4

So, the derivative of f(x) = 1/(5x + 1)^3 is f'(x) = -5/(5x + 1)^4.

2. f(x) = (9x + 5)^4

Step 1: Apply the power rule to the expression (9x + 5)^4: d/dx [(9x + 5)^4] = 4(9x + 5)^(4-1) * d/dx (9x + 5) = 4(9x + 5)^3 * 9 = 36(9x + 5)^3

Step 2: Now, we have the derivative of f(x) = (9x + 5)^4: f'(x) = d/dx [(9x + 5)^4] = 36(9x + 5)^3

So, the derivative of f(x) = (9x + 5)^4 is f'(x) = 36(9x + 5)^3.