Find the derivatives of the functions. y=(1)/(x) sin^-5 x-(x)/(3) cos^3 x

Name: Problem 30, Chapter 3: University Calculus: Early Transcendentals
Uploaded: 2020-05-03T19:59:27-08:00
Duration: 2 min 49 s
Channel: Robert Daugherty
Description: Find the derivatives of the functions. y=(1)/(x) sin^-5 x-(x)/(3) cos^3 x

step 1 Section 3 .6 problem number 30, we're asked to take derivatives where we're using the chain rule

Find the derivatives of the functions. y=(1)/(x) sin^-5...

Key Concepts

Differentiation of Trigonometric Functions

Differentiating trigonometric functions involves applying established rules for functions such as sine and cosine. The standard derivatives, such as the derivative of sin(x) being cos(x) and that of cos(x) being -sin(x), are often used in conjunction with other rules, like the chain rule, when these functions are composed with other functions or raised to a power.

Chain Rule

The chain rule is essential when dealing with composite functions, where one function is nested within another. It allows the differentiation of a composite function by stating that the derivative is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function.

Quotient Rule

The quotient rule is used to differentiate functions that are expressed as the quotient of two functions. It states that if a function is expressed as f(x)/g(x), its derivative is [f’(x)*g(x) - f(x)*g’(x)] divided by [g(x)]^2, providing a systematic approach to differentiating ratios.

Product Rule

The product rule is a method used to differentiate functions that are expressed as the product of two functions. According to this rule, if a function is the product of f(x) and g(x), then its derivative is given by f’(x)*g(x) + f(x)*g’(x), allowing the differentiation of more complex expressions that are multiplied together.

Power Rule

The power rule is a basic technique used to differentiate functions of the form x^n, where n is any real number. It states that the derivative of x raised to the power n is n*x^(n-1), making it especially useful when dealing with polynomial expressions or functions with power expressions.

Derivative

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. It represents the instantaneous rate of change or the slope of the tangent line at any given point on a function's graph.

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