Derivatives

Chain Rule and Higher Derivatives

Chain Rule

THEOREM

Chain Rule

If $g$ is differentiable at $x$ and $f$ is differentiable at $g(x)$ ,

(f(g(x)))'=f'(g(x))\cdot g'(x)

\displaystyle y=\sin(1x^2)

Inner coefficient b1Input x1

\displaystyle u=bx^2,\quad \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}=\cos(bx^2)\,(2bx)

\displaystyle u=1,\quad \frac{dy}{du}=0.540302,\quad \frac{du}{dx}=2

\displaystyle \frac{dy}{dx}=1.080605

Find the derivative of the following functions

(a) $\displaystyle f(x)=\cos(x^3)$

(b) $\displaystyle f(x)=(x^2+3x+2)^5$

(d) $\displaystyle f(x)=e^{\sec(x)}$

(e) $\displaystyle f(x)=\sin^3\left(x^2+\dfrac1x\right)$

Higher Order Derivatives

What happens if we take the derivative of a derivative—the second derivative? What if we do this many times?

Second derivative notations:

\frac{d^2y}{dx^2}\qquad y''\qquad f''(x)

Third derivative notations:

\frac{d^3y}{dx^3}\qquad y'''\qquad f'''(x)

Higher order derivative notations:

\frac{d^ny}{dx^n}\qquad y^{(n)}\qquad f^{(n)}(x)

Derivative order0

\displaystyle f(x)=2x^4

Input x1

\displaystyle f^{(0)}(1)=2

The fourth derivative is constant, and the fifth derivative is zero.

QUESTION

What is the derivative of

\displaystyle \cos(x^3)

Let $\displaystyle f(x) = 2x^4$ , find $\displaystyle f'(x)$ , $\displaystyle f''(x)$ , $\displaystyle f'''(x)$ , $\displaystyle f^{(4)}(x)$ , $\displaystyle f^{(5)}(x)$

◀ Previous SectionCombining Derivative Rules Next Section ▶Implicit Differentiation

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Derivatives

Chain Rule and Higher Derivatives

Chain Rule

THEOREM

Chain Rule

If $g$ is differentiable at $x$ and $f$ is differentiable at $g(x)$ ,

(f(g(x)))'=f'(g(x))\cdot g'(x)

\displaystyle y=\sin(1x^2)

Inner coefficient b1Input x1

\displaystyle u=bx^2,\quad \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}=\cos(bx^2)\,(2bx)

\displaystyle u=1,\quad \frac{dy}{du}=0.540302,\quad \frac{du}{dx}=2

\displaystyle \frac{dy}{dx}=1.080605

Find the derivative of the following functions

(a) $\displaystyle f(x)=\cos(x^3)$

(b) $\displaystyle f(x)=(x^2+3x+2)^5$

(d) $\displaystyle f(x)=e^{\sec(x)}$

(e) $\displaystyle f(x)=\sin^3\left(x^2+\dfrac1x\right)$

Higher Order Derivatives

What happens if we take the derivative of a derivative—the second derivative? What if we do this many times?

Second derivative notations:

\frac{d^2y}{dx^2}\qquad y''\qquad f''(x)

Third derivative notations:

\frac{d^3y}{dx^3}\qquad y'''\qquad f'''(x)

Higher order derivative notations:

\frac{d^ny}{dx^n}\qquad y^{(n)}\qquad f^{(n)}(x)

Derivative order0

\displaystyle f(x)=2x^4

Input x1

\displaystyle f^{(0)}(1)=2

The fourth derivative is constant, and the fifth derivative is zero.

QUESTION

What is the derivative of

\displaystyle \cos(x^3)

Let $\displaystyle f(x) = 2x^4$ , find $\displaystyle f'(x)$ , $\displaystyle f''(x)$ , $\displaystyle f'''(x)$ , $\displaystyle f^{(4)}(x)$ , $\displaystyle f^{(5)}(x)$

◀ Previous SectionCombining Derivative Rules Next Section ▶Implicit Differentiation

Chain Rule and Higher Derivatives

Chain Rule

Chain Rule

Multiply the outer and inner rates

Differentiate five compositions

Higher Order Derivatives

Take one derivative at a time

KNOWLEDGE QUIZ

Differentiate repeatedly

Chain Rule and Higher Derivatives

Chain Rule

Chain Rule

Multiply the outer and inner rates

Differentiate five compositions

Higher Order Derivatives

Take one derivative at a time

KNOWLEDGE QUIZ

Differentiate repeatedly