Answer to Question #273856 in Calculus for Annah

Question

^{Find the derivative of the following}

^{1.y= (2x²+6)⁵}

2. y=(5x-1/2x+3)³
      

3.g(x)=3xe^3x

4. y=e^cosx

Answer 1

1) $\big((2x^2+6)^5\big)^\prime =5\cdot (2x^2+6)^4\cdot (2x^2+6)^\prime =5\cdot (2x^2+6)^4\cdot 4x=20x(2x^2+6)^4$

2) $\bigg(\big(\frac{5x-1}{2x+3}\big)^3\bigg)^\prime =3\cdot \big(\frac{5x-1}{2x+3}\big)^2 \cdot \big(\frac{5x-1}{2x+3}\big)^\prime= 3\cdot \big(\frac{5x-1}{2x+3}\big)^2 \cdot \frac{5(2x+3)-2(5x-1)}{(2x+3)^2}=51\cdot \frac{(5x-1)^2}{(2x+3)^4}$

3) $(3xe^{3x})^\prime =(3x)^\prime \cdot e^{3x}+3x\cdot (e^{3x})^\prime =3e^{3x}+3x\cdot e^{3x}\cdot3=9xe^{3x}+3e^{3x}$

4) $(e^{cos x})^\prime =e^{cos x}\cdot (\cos x)^\prime=-\sin x \cdot e^{\cos x}$