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3.7: The Chain Rule
https://math.libretexts.org/Courses/City_University_of_New_York/Calculus_I_(CUNY)/03%3A_Derivatives/3.07%3A_The_Chain_Rule
Key Concepts The chain rule allows us to differentiate compositions of two or more functions. It states that for $h(x)=f(g(x)),$ $h′(x)=f′(g(x))g′(x).$ We can use the chain rule with other rules t...Key Concepts The chain rule allows us to differentiate compositions of two or more functions. It states that for $h(x)=f(g(x)),$ $h′(x)=f′(g(x))g′(x).$ We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them. The chain rule combines with the power rule to form a new rule: If $h(x)=(g(x))^n$ ,then $h′(x)=n(g(x))^{n−1}g′(x)$ .
3.8: The Chain Rule
https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT301_Calculus_I/03%3A_Derivatives/3.08%3A_The_Chain_Rule
Thus, if we think of $h(x)=\sin(x^3)$ as the composition $(f∘g)(x)=f(g(x))$ where $f(x)= \sin x$ and $g(x)=x^3$ , then the derivative of $h(x)=\sin(x^3)$ is the product of the derivative of \...Thus, if we think of $h(x)=\sin(x^3)$ as the composition $(f∘g)(x)=f(g(x))$ where $f(x)= \sin x$ and $g(x)=x^3$ , then the derivative of $h(x)=\sin(x^3)$ is the product of the derivative of $g(x)=x^3$ and the derivative of the function $f(x)=\sin x$ evaluated at the function $g(x)=x^3$ .
3.6: The Chain Rule
https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_(Hutchinson)/03%3A_Differentiation/3.06%3A_The_Chain_Rule
Thus, if we think of $h(x)=\sin(x^3)$ as the composition $(f∘g)(x)=f\big(g(x)\big)$ where $f(x)= \sin x$ and $g(x)=x^3$ , then the derivative of $h(x)=\sin(x^3)$ is the product of the derivat...Thus, if we think of $h(x)=\sin(x^3)$ as the composition $(f∘g)(x)=f\big(g(x)\big)$ where $f(x)= \sin x$ and $g(x)=x^3$ , then the derivative of $h(x)=\sin(x^3)$ is the product of the derivative of $g(x)=x^3$ and the derivative of the function $f(x)=\sin x$ evaluated at the function $g(x)=x^3$ .
8.1: The Chain Rule
https://math.libretexts.org/Bookshelves/Calculus/Differential_Calculus_for_the_Life_Sciences_(Edelstein-Keshet)/08%3A_Introducing_the_Chain_Rule/8.01%3A_The_Chain_Rule
Informally, the chain rule states that the change in y with respect to x is a product of two rates of change: (1) the rate of change of y with respect to its immediate input u, and (2) the r...Informally, the chain rule states that the change in y with respect to x is a product of two rates of change: (1) the rate of change of y with respect to its immediate input u, and (2) the rate of change of u with respect to its input, x.
3.6: The Chain Rule
https://math.libretexts.org/Courses/Mission_College/Math_3A%3A_Calculus_1_(Sklar)/03%3A_Derivatives/3.06%3A_The_Chain_Rule
Key Concepts The chain rule allows us to differentiate compositions of two or more functions. It states that for $h(x)=f(g(x)),$ $h′(x)=f′(g(x))g′(x).$ We can use the chain rule with other rules t...Key Concepts The chain rule allows us to differentiate compositions of two or more functions. It states that for $h(x)=f(g(x)),$ $h′(x)=f′(g(x))g′(x).$ We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them. The chain rule combines with the power rule to form a new rule: If $h(x)=(g(x))^n$ ,then $h′(x)=n(g(x))^{n−1}g′(x)$ .
3.7: The Chain Rule
https://math.libretexts.org/Courses/Laney_College/Math_3A%3A_Calculus_1_(Fall_2022)/03%3A_Derivatives/3.07%3A_The_Chain_Rule
Key Concepts The chain rule allows us to differentiate compositions of two or more functions. It states that for $h(x)=f(g(x)),$ $h′(x)=f′(g(x))g′(x).$ We can use the chain rule with other rules t...Key Concepts The chain rule allows us to differentiate compositions of two or more functions. It states that for $h(x)=f(g(x)),$ $h′(x)=f′(g(x))g′(x).$ We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them. The chain rule combines with the power rule to form a new rule: If $h(x)=(g(x))^n$ ,then $h′(x)=n(g(x))^{n−1}g′(x)$ .
4.1: Definition and Basic Properties of the Derivative
https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/04%3A_Differentiation/4.01%3A_Definition_and_Basic_Properties_of_the_Derivative
For example, given functions $f: G_{1} \rightarrow \mathbb{R}$ , $g: G_{2} \rightarrow \mathbb{R}$ , and $h: G_{3} \rightarrow \mathbb{R}$ such that $f\left(G_{1}\right) \subset G_{2}$ , \(g\left...For example, given functions $f: G_{1} \rightarrow \mathbb{R}$ , $g: G_{2} \rightarrow \mathbb{R}$ , and $h: G_{3} \rightarrow \mathbb{R}$ such that $f\left(G_{1}\right) \subset G_{2}$ , $g\left(G_{2}\right) \subset G_{3}$ , $f$ is differentiable at $a$ , $g$ is differentiable at $f(a)$ , and $h$ is differentiable at $g(f(a))$ , we obtain that $h \circ g \circ f$ is differentiable at $a$ and $(h \circ g \circ f)^{\prime}(a)=h^{\prime}(g(f(a))) g^{\prime}(f(a)) f^{\prime}(a)$
2.5: The Chain Rule
https://math.libretexts.org/Under_Construction/Purgatory/Book%3A_Active_Calculus_(Boelkins_et_al.)/02%3A_Computing_Derivatives/2.05%3A_The_Chain_Rule
In this section, we encountered the following important ideas: A composite function is one where the input variable x first passes through one function, and then the resulting output passes through a...In this section, we encountered the following important ideas: A composite function is one where the input variable x first passes through one function, and then the resulting output passes through another.
3.6: The Chain Rule
https://math.libretexts.org/Courses/Mission_College/Math_3A%3A_Calculus_I_(Reed)/03%3A_Derivatives/3.06%3A_The_Chain_Rule
Key Concepts The chain rule allows us to differentiate compositions of two or more functions. It states that for $h(x)=f(g(x)),$ $h′(x)=f′(g(x))g′(x).$ We can use the chain rule with other rules t...Key Concepts The chain rule allows us to differentiate compositions of two or more functions. It states that for $h(x)=f(g(x)),$ $h′(x)=f′(g(x))g′(x).$ We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them. The chain rule combines with the power rule to form a new rule: If $h(x)=(g(x))^n$ ,then $h′(x)=n(g(x))^{n−1}g′(x)$ .
6.4: The Chain Rule
https://math.libretexts.org/Courses/Hope_College/Math_125%3A_Hope_College/06%3A_Differentiation_Rules_and_Applications/6.04%3A_The_Chain_Rule
In this section, we encountered the following important ideas: A composite function is one where the input variable x first passes through one function, and then the resulting output passes through a...In this section, we encountered the following important ideas: A composite function is one where the input variable x first passes through one function, and then the resulting output passes through another.
3.6: The Chain Rule
https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/03%3A_Derivatives/3.06%3A_The_Chain_Rule
Key Concepts The chain rule allows us to differentiate compositions of two or more functions. It states that for $h(x)=f(g(x)),$ $h′(x)=f′(g(x))g′(x).$ We can use the chain rule with other rules t...Key Concepts The chain rule allows us to differentiate compositions of two or more functions. It states that for $h(x)=f(g(x)),$ $h′(x)=f′(g(x))g′(x).$ We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them. The chain rule combines with the power rule to form a new rule: If $h(x)=(g(x))^n$ ,then $h′(x)=n(g(x))^{n−1}g′(x)$ .

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