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3.4

The Chain Rule

The Chain Rule

$F\text{'}\left(x\right) = f\text{'}\left(g\right(x\left)\right)\times g\text{'}\left(x\right)$

if g is differentiable at x and f is differentiable at g(x), then the composite function F = f * g defined by F(x) = f(g(x)) is differenciable at x and F' is given by the product above.

The Chain Rule (Leibniz notation)

$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$

Same Chain Rule as above, but written in Leibniz notation. This is true if y = f(u) and u=g(x) are both differentiable functions.

The Chain Rule Combined With the Power Rule

$$\begin{gathered} \frac{d}{dx}\left(u^n\right) = n{u}^{n-1}\frac{du}{dx} \\ \text{or: } \\ \frac{d}{dx}\left[g\right(x){]}^n=n[g\left(x\right){]}^{n-1}\times g\text{'}\left(x\right) \end{gathered}$$

if n is and real number and u = g(x) is differentiable. This also means that $\frac{d}{dx}\left(a^x\right) = a^x\ln a$

Links

• Home
• 3.1: Derivatives of Polynomials and Exponential Functions
• 3.2: The Product and Quotient Rules
• 3.3: Derivatives of Trigonometric Functions
• 3.4: The Chain Rule
• 3.5: Implicit Differentiation
• 3.6: Steps in Logarithmic Differentiation
• 3.7: Rates of Change in the Natural and Social Sciences
• 3.8: Exponential Growth and Decay
• 3.9: Related Rates
• 3.10: Linear Approximations and Differentials
• 3.11: Hyperbolic Functions