3. Techniques of Differentiation

27–76. Calculate the derivative of the following functions.

$y={(x^2+2x+7)}^8$

27–76. Calculate the derivative of the following functions.

$y=\sqrt{10x+1}$

27–76. Calculate the derivative of the following functions.

$\sqrt[3]{x^2+9}$

27–76. Calculate the derivative of the following functions.

$y={(3x^2+7x)}^{10}$

Calculate the derivative of the following functions.

y = (p+3)² sin p²

Calculate the derivative of the following functions.

y = e^2x(2x-7)⁵

Calculate the derivative of the following functions.

y = (e^x / x+1)⁸

Calculate the derivative of the following functions.

y = √x+√x+√x

Calculate the derivative of the following functions.

y = (f(g(x^m)))^n, where f and g are differentiable for all real numbers and m and n are constants

Derivatives by different methods

a. Calculate d/dx (x²+x)² using the Chain Rule. Simplify your answer.

Second derivatives Find d²y/dx²for the following functions.

y = e^-2x²

Second derivatives Find d²y/dx²for the following functions.

y = √x²+2

Second derivatives Find d²y/dx²for the following functions.

y = x cos x²

Vibrations of a spring Suppose an object of mass m is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position $y=0$ when the mass hangs at rest. Suppose you push the mass to a position $y_0$ units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system), the position y of the mass after t seconds is $y=y_0\cos \left(t\sqrt{\frac{k}{m}}\right)$, where $k>0$ is a constant measuring the stiffness of the spring (the larger the value of $k$, the stiffer the spring) and $y$ is positive in the upward direction.

Use equation (4) to answer the following questions.

c. How would the velocity be affected if the experiment were repeated with a spring having four times the stiffness ($k$ is increased by a factor of $4$)?

A general proof of the Chain Rule Let f and g be differentiable functions with h(x)=f(g(x)). For a given constant a, let u=g(a) and v=g(x), and define H (v) = <1x1 matrix>

c. Show that h′(a) = lim x→a ((H(g(x))+f′(g(a)))⋅g(x)−g(a)/x−a).