Question

Помогите пожалуйста найти производную функцию

Answer 1

1)

$f(x) = x^2-15\\\\f'(x) = \left(x^2\right)' - (15)'\\\\f'(x) = 2x - 0\\\\\boxed{\bf{f'(x) = 2x}}$

2)

$f(x) = \left(x+1\right)^3\\\\f'(x) = \left(x+1\right)'\cdot3(x+1)^2\\\\f'(x) = \left((x)' + (1)'\right)\cdot 3(x+1)^2\\\\f'(x) = (1 + 0)\cdot 3(x+1)^2\\\\\boxed{\bf{f'(x) = 3(x+1)^2}}$

3)

$f(x) = 3x + 41 + x^2 + x^3\\\\f'(x) = 3\cdot (x') + (41)' + \left(x^2\right)' + \left(x^3\right)'\\\\f'(x) = 3\cdot 1 + 0 + 2x + 3x^2\\\\\boxed{\bf{f'(x) = 3 + 2x + 3x^2}}$

4)

$f(x) = \dfrac{x^3}{4-x}\\\\\\f'(x) = \dfrac{\left(x^3\right)'\cdot (4-x) - x^3\cdot (4-x)'}{\left(4-x\right)^2}\\\\\\f'(x) = \dfrac{3x^2\cdot (4-x)-x^3\cdot\left((4)' - (x)'\right)}{\left(4-x\right)^2}\\\\\\f'(x) = \dfrac{3x^2(4-x) -x^3\cdot (0 - 1)}{\left(4-x\right)^2}\\\\\\f'(x) = \dfrac{12x^2 - 3x^3 + x^3}{\left(4-x\right)^2}\\\\\\f'(x) = \dfrac{12x^2 - 2x^3}{\left(4-x\right)^2}\\\\\\\boxed{\bf{f'(x) = \dfrac{2x^2(6-x)}{\left(4-x\right)^2}}}$

5)

$f(x) = -\dfrac{1}{2}\cdot \cos4x\\\\\\f'(x) = \left(-\dfrac{1}{2}\right)'\cdot \cos4x + \left(-\dfrac{1}{2}\right)\cdot(\cos4x)'\\\\\\f'(x) = 0\cdot \cos4x - \dfrac{1}{2}\cdot (4x)'\cdot (-\sin4x)\\\\\\f'(x)= \dfrac{1}{2}\cdot 4\cdot\sin4x\\\\\boxed{\bf{f'(x) = 2\sin4x}}$

6)

$f(x) = \dfrac{1}{3}\cdot \sin x\\\\\\f'(x) = \left(\dfrac{1}{3}\right)'\cdot\sin x + \dfrac{1}{3}\cdot(\sin x)'\\\\\\f'(x) = 0\cdot\sin x + \dfrac{1}{3}\cdot \cos x\\\\\\\boxed{\bf{f'(x) = \dfrac{\cos x}{3}}}$

7)

$f(x) = 2 + \sqrt{x}\\\\f'(x) = (2)' + \left(\sqrt{x}\right)'\\\\\\f'(x) = 0 + \dfrac{1}{2\sqrt{x}}\\\\\\\boxed{\bf{f'(x) = \dfrac{1}{2\sqrt{x}}}}$