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Zorluk: KolayRules of Differentiation (Product, Quotient, and Chain Rules)

If $y = (2x + 1)^3$ , find the value of $\frac{dy}{dx}$ at $x = 1$ .

Cevap: 54

Cevap

54

Applying the chain rule gives

\frac{dy}{dx} = 3(2x + 1)^2 \cdot 2 = 6(2x + 1)^2

. Evaluating at

x = 1

gives

6(3)^2 = 54

.

Adım Adım Çözüm

1

Differentiate

y = (2x + 1)^3

using the chain rule.

\frac{dy}{dx} = 6(2x + 1)^2

According to the chain rule,

\frac{d}{dx}[u^n] = n u^{n-1} \cdot \frac{du}{dx}

, where

u = 2x + 1

and

\frac{du}{dx} = 2

.

2

Evaluate the derivative at

x = 1

.

\frac{dy}{dx}\Big|_{x=1} = 54

Substituting

x = 1

into

6(2x + 1)^2

yields

6(3)^2 = 54

.

Anahtar Kavram

Chain Rule for Differentiation

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Konular

Rules of Differentiation (Product, Quotient, and Chain Rules)

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