Find the derivatives of the two functions

Oct 13, 2025 by windailys.com 81 views

Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to one of its variables. It is a fundamental concept in mathematics and has numerous applications in various fields, including physics, engineering, and economics. In this article, we will discuss how to find the derivative of the function $y=(x+1)^5(x+4)^3$ .

Understanding the Function

The given function is a product of two functions, $(x+1)^5$ and $(x+4)^3$ . To find the derivative of this function, we will use the product rule of differentiation, which states that if $y=u(x)v(x)$ , then $y'=u'(x)v(x)+u(x)v'(x)$ .

Applying the Product Rule

To apply the product rule, we need to find the derivatives of the two functions, $(x+1)^5$ and $(x+4)^3$ . We can use the chain rule of differentiation to find these derivatives.

Derivative of $(x+1)^5$

The derivative of $(x+1)^5$ can be found using the chain rule. Let $u=x+1$ , then $y=u^5$ . The derivative of $y$ with respect to $u$ is $y'=5u^4$ . The derivative of $u$ with respect to $x$ is $u'=1$ . Therefore, the derivative of $(x+1)^5$ is $5(x+1)^4$ .

Derivative of $(x+4)^3$

The derivative of $(x+4)^3$ can also be found using the chain rule. Let $u=x+4$ , then $y=u^3$ . The derivative of $y$ with respect to $u$ is $y'=3u^2$ . The derivative of $u$ with respect to $x$ is $u'=1$ . Therefore, the derivative of $(x+4)^3$ is $3(x+4)^2$ .

Finding the Derivative of the Function

Now that we have found the derivatives of the two functions, we can apply the product rule to find the derivative of the given function.

\frac{dy}{dx} = 5(x+1)^4(x+4)^3 + (x+1)^5(3(x+4)^2)

Simplifying the Derivative

We can simplify the derivative by combining like terms.

\frac{dy}{dx} = 5(x+1)^4(x+4)^3 + 3(x+1)^5(x+4)^2

Final Answer

The final answer is $\boxed{5(x+1)^4(x+4)^3 + 3(x+1)^5(x+4)^2}$ .

Conclusion

In this article, we discussed how to find the derivative of the function $y=(x+1)^5(x+4)^3$ . We used the product rule of differentiation and the chain rule to find the derivatives of the two functions, and then applied the product rule to find the derivative of the given function. The final answer is $\boxed{5(x+1)^4(x+4)^3 + 3(x+1)^5(x+4)^2}$ .

Step-by-Step Solution

Here is the step-by-step solution to the problem:

Understand the function: The given function is a product of two functions, $(x+1)^5$ and $(x+4)^3$ .
Apply the product rule: The product rule states that if $y=u(x)v(x)$ , then $y'=u'(x)v(x)+u(x)v'(x)$ .
Find the derivatives of the two functions: We used the chain rule to find the derivatives of the two functions.
Apply the product rule to find the derivative of the given function: We applied the product rule to find the derivative of the given function.
Simplify the derivative: We combined like terms to simplify the derivative.

Frequently Asked Questions

What is the derivative of the function $y=(x+1)^5(x+4)^3$ ? The derivative of the function is $\boxed{5(x+1)^4(x+4)^3 + 3(x+1)^5(x+4)^2}$ .
How do I find the derivative of a product of two functions? You can use the product rule of differentiation, which states that if $y=u(x)v(x)$ , then $y'=u'(x)v(x)+u(x)v'(x)$ .
What is the chain rule of differentiation? The chain rule is a rule of differentiation that states that if $y=f(u)$ and $u=g(x)$ , then $y'=f'(u)g'(x)$ .

References

Calculus by Michael Spivak: This book provides a comprehensive introduction to calculus, including the product rule and the chain rule.
Calculus by James Stewart: This book provides a comprehensive introduction to calculus, including the product rule and the chain rule.
Derivatives by Khan Academy: This website provides a comprehensive introduction to derivatives, including the product rule and the chain rule.

Introduction

In the previous article, we discussed how to find the derivative of the function $y=(x+1)^5(x+4)^3$ . In this article, we will answer some frequently asked questions about derivatives of functions.

Q&A

Q: What is the derivative of the function $y=x^2$ ?

A: The derivative of the function $y=x^2$ is $\frac{dy}{dx}=2x$ .

Q: How do I find the derivative of a function that involves a trigonometric function?

A: To find the derivative of a function that involves a trigonometric function, you can use the chain rule of differentiation. For example, if $y=\sin(x^2)$ , then $\frac{dy}{dx}=\cos(x^2)(2x)$ .

Q: What is the derivative of the function $y=e^x$ ?

A: The derivative of the function $y=e^x$ is $\frac{dy}{dx}=e^x$ .

Q: How do I find the derivative of a function that involves a logarithmic function?

A: To find the derivative of a function that involves a logarithmic function, you can use the chain rule of differentiation. For example, if $y=\ln(x^2)$ , then $\frac{dy}{dx}=\frac{1}{x^2}(2x)$ .

Q: What is the derivative of the function $y=\sin(x)$ ?

A: The derivative of the function $y=\sin(x)$ is $\frac{dy}{dx}=\cos(x)$ .

Q: How do I find the derivative of a function that involves a polynomial function?

A: To find the derivative of a function that involves a polynomial function, you can use the power rule of differentiation. For example, if $y=x^3$ , then $\frac{dy}{dx}=3x^2$ .

Q: What is the derivative of the function $y=\cos(x)$ ?

A: The derivative of the function $y=\cos(x)$ is $\frac{dy}{dx}=-\sin(x)$ .

Q: How do I find the derivative of a function that involves a rational function?

A: To find the derivative of a function that involves a rational function, you can use the quotient rule of differentiation. For example, if $y=\frac{x^2}{x^3}$ , then $\frac{dy}{dx}=\frac{x^3(2x)-x^2(3x^2)}{(x^3)^2}$ .