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Chapter 3: Problem 41

Use the Chain Rule to find the derivative of the following functions. $$y=\left(2 x^{6}-3 x^{3}+3\right)^{25}$$

Short Answer

Expert verified
Answer: The derivative of the given function is $$\frac{dy}{dx} = 25(2x^6 - 3x^3 + 3)^{24} \cdot (12x^5 - 9x^2)$$.

Step by step solution

01

Identifying the inner and outer functions

The given function, $$y=\left(2x^6 - 3x^3 + 3\right)^{25}$$, is a composition of two functions. The outer function is $$f(u)=u^{25}$$ and the inner function is $$g(x)=2x^6 - 3x^3 + 3$$. The Chain Rule states that if we have a function in the form $$y = f(g(x))$$, then the derivative is $$\frac{dy}{dx} = \frac{df}{du}\cdot\frac{du}{dx}$$.
02

Computing the derivative of the outer function with respect to u

To find the derivative of $$y=u^{25}$$ with respect to u, we use the power rule for differentiation, which states that the derivative of $$u^n$$ with respect to u is $$nu^{n-1}$$. In this case, $$\frac{df}{du} = 25u^{24}$$.
03

Computing the derivative of the inner function with respect to x

Next, we find the derivative of $$g(x)=2x^6 - 3x^3 + 3$$ with respect to x. Using the power rule for each term, we get $$\frac{dg}{dx} = 12x^5 - 9x^2$$.
04

Applying the Chain Rule

Now we can apply the Chain Rule: $$\frac{dy}{dx} = \frac{df}{du}\cdot\frac{dg}{dx}$$, which means $$\frac{dy}{dx} = 25u^{24}\cdot(12x^5 - 9x^2)$$. Don't forget that we need to replace u with the inner function, so $$u=g(x)=2x^6-3x^3+3$$.
05

Simplify the derivative

Lastly, we plug in the expression for $$g(x)$$ back into the derivative formula for u and simplify: $$\frac{dy}{dx} = 25(2x^6 - 3x^3 + 3)^{24} \cdot (12x^5 - 9x^2)$$. Thus, the derivative of the given function is $$\frac{dy}{dx} = 25(2x^6 - 3x^3 + 3)^{24} \cdot (12x^5 - 9x^2)$$.

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Key Concepts

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

Derivative
The concept of a derivative is central to calculus. It measures how a function changes as its input changes. In simple terms, the derivative of a function at a certain point describes the slope of the tangent line at that point. This allows us to understand the rate of change or how one quantity changes in relation to another.
When finding the derivative, we're essentially calculating how sensitive our function is to changes in its input. In calculus, this is represented by \( \frac{dy}{dx} \), where \( y \) is the function and \( x \) is the variable.
Power Rule
The Power Rule is a fundamental differentiation technique that is easy to apply. If you have a function of the form \( x^n \) (where \( n \) is a constant), the derivative is simply \( nx^{n-1} \). This rule simplifies the process of finding derivatives of polynomial functions.
For example, if you have \( x^3 \), applying the Power Rule gives \( 3x^2 \).
This straightforward method is essential, especially when differentiating more complex functions that involve polynomials.
Composition of Functions
A composition of functions occurs when one function is applied to the result of another function. For example, if you have \( y = f(g(x)) \), \( g(x) \) is the inner function and \( f(u) \) is the outer function, where \( u = g(x) \).
This composition requires a particular approach when finding derivatives, known as the Chain Rule. Understanding how to deal with composed functions is crucial since many functions encountered in real-world applications are compositions of simpler functions.
Differentiation Techniques
Differentiation techniques include several methods to find the derivative of a function. One key technique is the Chain Rule, which is used when dealing with composed functions. The Chain Rule states:
  • If \( y = f(g(x)) \), then the derivative is \( \frac{dy}{dx} = \frac{df}{du} \cdot \frac{dg}{dx} \).

To apply this rule, first differentiate the outer function concerning its inner function, then multiply by the derivative of the inner function itself.
Knowing various differentiation techniques, such as the Power Rule and the Chain Rule, allows for solving more complicated calculus problems efficiently.

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Most popular questions from this chapter

Find the following higher-order derivatives. $$\frac{d^{3}}{d x^{3}}\left(x^{2} \ln x\right)$$.

Calculate the derivative of the following functions (i) using the fact that \(b^{x}=e^{x \ln b}\) and (ii) by using logarithmic differentiation. Verify that both answers are the same. $$y=\left(x^{2}+1\right)^{x}$$

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas \(y=c x^{2}\) form orthogonal trajectories with the family of ellipses \(x^{2}+2 y^{2}=k,\) where \(c\) and \(k\) are constants (see figure). Find \(d y / d x\) for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. \(y=c x^{2} ; x^{2}+2 y^{2}=k,\) where \(c\) and \(k\) are constants

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