Derivatives of Functions of f(x) = u 

Considering f(x) = (x + x^{2})^{5}(1 + x^{3})^{2}, find f'(x) Let’s rewrite the function as follows: y = (x^{2 }+ x)^{5}(1 + x^{3})^{2} Let’s consider a function k as follows: k = x^{2} + x k' = 2x + 1 Let’s consider a function h as follows: u = k^{5} = (x^{2} + x)^{5} u' = nk^{n1}k' = 5(x^{2} + x)^{4}(2x + 1) = 5(2x + 1)(x^{2} + x)^{4} Let’s consider a function g as follows: g = 1 + x^{3} g’ = 3x^{2} Let’s consider a function m as follows: v = g^{2} = (1 + x^{3})^{2} v' = ng^{n1}g' = 2(1 + x^{3})(3x^{2}) = 6x^{2}(1 + x^{3}) Let’s consider a function f as follows: f = uv To calculate the derivative, we use the formula (uv)' = u'v + uv': f' = u'v + uv' = 5(2x + 1)(x^{2} + x)^{4}(1 + x^{3})^{2} +(x^{2} + x)^{5}6x^{2}(1 + x^{3}) = 5(2x + 1)(x^{2} + x)^{4}(1 + x^{3})^{2} +6x^{2}(1 + x^{3})(x^{2} + x)^{5} = (x^{2} + x)^{4}(1 + x^{3})[5(2x + 1)(1 + x^{3}) + 6x^{2}(x^{2} + x)]
Let’s consider a function k as follows: g = 3x^{3} + 5 g' = 9x^{2} Let’s consider a function h as follows: u = g^{2} = (3x^{3} + 5)^{2} u' = ng^{n1}g' = 3(3x^{3} + 5)(9x^{2}) = 27x^{2}(3x^{3} + 5) Let’s consider a function k as follows: k = 4x^{2} + 2x + 3 k’ = 8x + 2 Let’s consider a function h as follows: v = k^{4} = (4x^{2} + 2x + 3)^{4} v’ = nk^{n1}k' = 4(4x^{2} + 2x + 3)^{3}(8x + 2) = 4(8x + 2)(4x^{2} + 2x + 3)^{3} Let’s consider a function f as follows: f = u/v = (3x^{3} + 5)^{2}/(4x^{2} + 2x + 3)^{4} f' = (u'v + uv')/v^{8} Considering f(x) = [3x + (2x + x^{5})^{2}]^{4}, find f'(x) Let’s rewrite the function as follows: y = (3x + (x^{5} + 2x)^{2})^{4} Let’s consider a function k as follows: k = x^{5} + 2x k' = 5x^{4} + 2 Let’s consider a function h as follows: h = k^{2} = (x^{5} + 2x)^{2} h' = nk^{n1}k' = 2( x^{5} + 2x)^{3}(5x^{4} + 2) = 2(5x^{4} + 2)(x^{5} + 2x)^{3} Let’s consider a function g as follows: g = 3x + h = 3x + (x^{5} + 2x)^{2} g’ = (3x)' + h' = 3 + (2(5x^{4} + 2)(x^{5} + 2x)^{3}) = 2(5x^{4} + 2)(x^{5} + 2x)^{3} + 3 Let’s consider a function f as follows: f = g^{4} = [3x + (x^{5} + 2x)^{2}]^{4} f' = ng^{n1}g' = 4[3x + (x^{5} + 2x)^{2}]^{5}[2(5x^{4} + 2)(x^{5} + 2x)^{3} + 3] = 4[2(5x^{4} + 2)(x^{5} + 2x)^{3} + 3][3x + (x^{5} + 2x)^{2}]^{5}



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