3. Techniques of Differentiation

Calculate the derivative of the following functions.

y = csc 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 = e^√x

Calculate the derivative of the following functions.

y = tan(xe^x)

Find the derivative of the function.

$y=(8x^3-2x{)}^{3/2}$

Find the derivatives of the functions in Exercises 1–42.

s = (sec t + tan t)⁵

Find the derivative of the function.

$f\left(t\right)=(3t^2+7t-2{)}^{10}$

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 = (3x+7)¹⁰

Find the derivatives of the functions in Exercises 19–40.

p = √(3 − t)

Deriving trigonometric identities

a. Differentiate both sides of the identity cos 2t = cos² t−sin² t to prove that sin 2 t= 2 sin t cos t.

Calculate the derivative of the following functions.

y = sin(sin(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 = sin x⁵

Second derivatives Find d²y/dx²for the following functions.

y = x cos x²

{Use of Tech} Tangent line Find the equation of the line tangent to y=2^sin x at x=π/2. Graph the function and the tangent line.

Calculate the derivative of the following functions.

y = ⁴√(2x / (4x - 3))

Calculate the derivative of the following functions.

y = (p+3)² sin p²