3. Techniques of Differentiation

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.

y = √7x-1

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.

y = e^4x²+1

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.

y = e^√x

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.

y = tan 5x²

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