Derivatives in Differential Form


In Exercises 17–28, find dy.


y = sin(5√x)

Verified step by step guidance

1. First, identify the function y = sin(5√x). We need to find the derivative dy/dx.
2. Recognize that the function involves a composition of functions: the sine function and the square root function. Use the chain rule to differentiate.
3. The chain rule states that if you have a composite function y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). Here, f(u) = sin(u) and g(x) = 5√x.
4. Differentiate the outer function f(u) = sin(u) with respect to u, which gives f'(u) = cos(u).
5. Differentiate the inner function g(x) = 5√x with respect to x. Recall that √x can be expressed as x^(1/2), so g(x) = 5x^(1/2). The derivative g'(x) = 5 * (1/2) * x^(-1/2).

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative dy/dx is found by multiplying the derivative of the outer function f with respect to the inner function g by the derivative of the inner function g with respect to x. This is essential for differentiating y = sin(5√x), where the outer function is sine and the inner function is 5√x.

Derivative of Sine Function

The derivative of the sine function, sin(x), is cos(x). This basic derivative rule is crucial when applying the chain rule to differentiate y = sin(5√x). After applying the chain rule, the derivative of the outer function, sin, is needed, which transforms into cos when differentiating.

Derivative of Square Root Function

The derivative of the square root function, √x, is 1/(2√x). This rule is important when differentiating the inner function 5√x in the given problem. By applying this derivative rule, we can find the rate of change of the inner function, which is necessary for the application of the chain rule.