Derivatives

Combining Derivative Rules

What about combining these basic derivative rules? How do we differentiate sums, differences, products, quotients, or constant multiples of functions? For the following rules, assume $f$ and $g$ are differentiable at the points in question.

Multiplication by a Constant

THEOREM

Multiplication by a Constant

(kf(x))'=kf'(x)

Find the derivative of $\displaystyle f(x) = 5\ln(x)$

Sum / Difference

THEOREM

Sum / Difference

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

Find the derivative of $\displaystyle f(x) = \ln(x) + 4x^3 - 6\csc(x)$

Product Rule

THEOREM

Product Rule

(fg)'(x)=f'(x)g(x)+f(x)g'(x)

Find the derivative of $\displaystyle f(x) = \ln(x)\cos(x)$

Quotient Rule

THEOREM

Quotient Rule

For $g(x)\ne0$ ,

\left(\frac fg\right)'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}

QUESTION

Which is the derivative of

\displaystyle \ln(x)\cos(x)

for

\displaystyle x>0

Find the derivative of $\displaystyle f(x) = {\dfrac{5^x}{x^2}}$

Use the Quotient Rule to prove that $\displaystyle {\dfrac{d}{dx}}\cot(x) = -\csc^2(x)$

◀ Previous SectionBasic Derivative Rules Next Section ▶Chain Rule and Higher Derivatives

Loading lesson… / 正在打开教程…

Leafletv4.3.3

Courses Pro Get Pro Login / Register

Derivatives

Combining Derivative Rules

Multiplication by a Constant

THEOREM

Multiplication by a Constant

(kf(x))'=kf'(x)

Find the derivative of $\displaystyle f(x) = 5\ln(x)$

Sum / Difference

THEOREM

Sum / Difference

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

Find the derivative of $\displaystyle f(x) = \ln(x) + 4x^3 - 6\csc(x)$

Product Rule

THEOREM

Product Rule

(fg)'(x)=f'(x)g(x)+f(x)g'(x)

Find the derivative of $\displaystyle f(x) = \ln(x)\cos(x)$

Quotient Rule

THEOREM

Quotient Rule

For $g(x)\ne0$ ,

\left(\frac fg\right)'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}

QUESTION

Which is the derivative of

\displaystyle \ln(x)\cos(x)

for

\displaystyle x>0

Find the derivative of $\displaystyle f(x) = {\dfrac{5^x}{x^2}}$

Use the Quotient Rule to prove that $\displaystyle {\dfrac{d}{dx}}\cot(x) = -\csc^2(x)$

◀ Previous SectionBasic Derivative Rules Next Section ▶Chain Rule and Higher Derivatives

Combining Derivative Rules