Derivatives


In Exercises 1–18, find dy/dx.


f(x) = x³ sin x cos x

Verified step by step guidance

1. Identify the function f(x) = x³ sin(x) cos(x) and recognize that it is a product of three functions: u(x) = x³, v(x) = sin(x), and w(x) = cos(x).
2. Apply the product rule for derivatives, which states that if you have a product of functions u(x), v(x), and w(x), then the derivative is given by: (uvw)' = u'vw + uv'w + uvw'.
3. Calculate the derivative of each individual function: u'(x) = 3x², v'(x) = cos(x), and w'(x) = -sin(x).
4. Substitute these derivatives into the product rule formula: dy/dx = (3x²)(sin(x))(cos(x)) + (x³)(cos(x))(cos(x)) + (x³)(sin(x))(-sin(x)).
5. Simplify the expression by combining like terms and using trigonometric identities if necessary to express the derivative in its simplest form.

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at any given point. The notation dy/dx indicates the derivative of y with respect to x, and it can be computed using various rules such as the power rule, product rule, and chain rule.

Product Rule

The product rule is a formula used to find the derivative of the product of two functions. If u(x) and v(x) are two differentiable functions, the product rule states that the derivative of their product is given by d(uv)/dx = u'v + uv', where u' and v' are the derivatives of u and v, respectively. This rule is essential for differentiating functions that are products of simpler functions, such as the given function f(x) = x³ sin x cos x.

Trigonometric Functions

Trigonometric functions, such as sine (sin) and cosine (cos), are fundamental in calculus, especially when dealing with derivatives. These functions have specific derivatives: the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). Understanding these derivatives is crucial when applying the product rule to functions that involve trigonometric components, as seen in the function f(x) = x³ sin x cos x.