Problem 5 Differentiate, with respect to \... [FREE SOLUTION]

Chapter 29: Problem 5

Differentiate, with respect to $x$, $$ y=5 x^{4}+4 x-\frac{1}{2 x^{2}}+\frac{1}{\sqrt{x}}-3 $$

Short Answer

Expert verified

The derivative is $y' = 20x^3 + 4 + \frac{1}{x^3} - \frac{1}{2}x^{-3/2}$.

Step by step solution

Apply the Power Rule

Start by differentiating the polynomial terms in the expression: $5x^4$ and $4x$. The power rule states that the derivative of $x^n$ is $nx^{n-1}$. Thus, the derivative of $5x^4$ is $20x^3$ and the derivative of $4x$ is $4$.

Differentiate the Rational Function

Next, differentiate the term $-\frac{1}{2x^2}$. Rewrite $-\frac{1}{2x^2}$ as $-\frac{1}{2}x^{-2}$ and apply the power rule. The derivative is $-\frac{1}{2}(-2)x^{-3} = x^{-3} = \frac{1}{x^3}$.

Differentiate the Square Root Function

Differentiate the term $\frac{1}{\sqrt{x}}$. Rewrite $\frac{1}{\sqrt{x}}$ as $x^{-1/2}$. Applying the power rule gives $-\frac{1}{2}x^{-3/2}$ as the derivative.

Differentiate the Constant

The derivative of a constant, $-3$, is $0$ because constants do not change and therefore have no rate of change.

Combine the Derivatives

Combine all the calculated derivatives. The derivative of $y$ with respect to $x$ is:\[ y' = 20x^3 + 4 + \frac{1}{x^3} - \frac{1}{2}x^{-3/2}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule

Differentiation is a fundamental concept in calculus used to find the rate at which a function changes at any point. When differentiating polynomial terms like $5x^4$ and $4x$, the power rule is a quick and easy method. The power rule states that if you have a term $x^n$, its derivative is $nx^{n-1}$. This means you multiply the exponent by the coefficient and reduce the exponent by one.

For $5x^4$, using the power rule, multiply 5 by 4 to get 20 and decrease the exponent 4 by 1, resulting in the derivative $20x^3$.
Similarly, the derivative of $4x$, which is equivalent to $4x^1$, is simply 4, as subtracting the exponent yields $4x^0 = 4$.

Try this rule on different polynomial terms to get a feel for how straightforward it is. It becomes second nature with practice!

Polynomial Differentiation

Polynomial differentiation extends the power rule across each term of a polynomial independently. In a polynomial like $5x^4 + 4x$, apply the power rule to each term one by one. Each term can be differentiated separately because differentiation is linear, meaning the derivative of a sum is the sum of the derivatives.
Applying the power rule to polynomials you encounter in your exercise is systematic and methodical:

Identify each term's exponent.
Apply the power rule to each term separately.
Combine the resulting derivatives into a single expression.

This approach ensures clarity and accuracy when working on more complex polynomial functions.

Rational Function Differentiation

Rational functions, such as terms like $-\frac{1}{2x^2}$, may appear tricky at first glance. The key is to rewrite these expressions using negative exponents to utilize the power rule effectively. For $-\frac{1}{2x^2}$, rewrite it as $-\frac{1}{2}x^{-2}$. Now, you can apply the power rule:

Multiply the exponent, $-2$, with the coefficient, $-\frac{1}{2}$, resulting in 1.
Decrease the exponent by 1 to get $-3$, resulting in the derivative $x^{-3}$.

Rewriting rational functions in this manner simplifies the differentiation process and helps to avoid mistakes.

Square Root Differentiation

Differentiating square roots transforms the problem into a more manageable form using negative fractional exponents. The term $\frac{1}{\sqrt{x}}$ is equivalent to $x^{-1/2}$.Apply the power rule to this new expression:

Multiply the exponent $-1/2$ by 1 (the coefficient in front), giving $-1/2$.
Decrease the original exponent by 1 to obtain $-3/2$.
The derivative is therefore $-\frac{1}{2}x^{-3/2}$.

By converting square roots into fractional exponents, differentiation becomes more straightforward, leveraging the power rule for efficient calculations.

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91影视

Differentiate, with respect to \(x\), $$ y=5 x^{4}+4 x-\frac{1}{2 x^{2}}+\frac{1}{\sqrt{x}}-3 $$

Short Answer

Step by step solution

Apply the Power Rule

Differentiate the Rational Function

Differentiate the Square Root Function

Differentiate the Constant

Combine the Derivatives

Key Concepts

Power Rule

Polynomial Differentiation

Rational Function Differentiation

Square Root Differentiation

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