Differentiate a product or quotient of two functions

Which first decision matches the displayed function?

Expand the square of x multiplied by the sum of x and a, and compare its derivative at one as a changes

Which conclusion fits every setting?

Product and quotient rules

A product or quotient can sometimes be simplified first. If you use these rules, differentiate u and v separately, then combine the contributions in the correct order

1. 1Name the structure Use the product rule for two factors multiplied together and the quotient rule for a numerator divided by a denominator \(\displaystyle f(x)=u(x)v(x)\quad\text{or}\quad f(x)=\frac{u(x)}{v(x)}\)
2. 2Differentiate each factor Find the derivative of each factor separately before combining the results \(\displaystyle u=2x+1,\quad v=x^2+3,\quad u'=2,\quad v'=2x\)
3. 3Use the product rule Keep one factor unchanged in each term and match each derivative with the other factor \(\displaystyle \frac{d}{dx}(uv)=u'v+uv'\)
4. 4Use the quotient rule For a quotient, put v u' minus u v' over the square of the denominator and keep v non-zero \(\displaystyle \frac{d}{dx}\left(\frac{u}{v}\right)=\frac{vu'-uv'}{v^2},\quad v\ne0\)

Given

\(\displaystyle u(x)=2x+1\)

Find u′(1)

Given

\(\displaystyle f'(1)=2\left(1^2+3\right)+\left(2(1)+1\right)\left(2(1)\right)\)

Evaluate the given expression for f′(1)

Which product-rule line is correct?

Construct a composite derivative

Identify the outer power and the inner quadratic. Differentiate the outer function while leaving the inner expression intact, then multiply by the derivative of the inner expression

Construct the full expression

Write a complete derivative in a factored form. Label the outer and inner derivatives and verify the result at x=1. Explain why the inner derivative cannot be omitted

1. 1Your written solution Write the complete rule, derivation and domain on paper. Written practice is ungraded

Task

\(\displaystyle g(x)=(2x^2-x+1)^3\)

Evaluate g′(1)

Review the whole expression

The inner derivative is 4x−1, and the outer derivative is three times the square of the unchanged inner expression. Both function and derivative have domain R.

1. 1Review criteria Check every term and sign, the original domain and any excluded inputs, then independently substitute or differentiate to verify. Explain discrepancies and correct your own written expression

Construct both product terms

Each factor is differentiated once while the other is retained. Factor only after both terms have been written; multiplying the two derivatives is not the product rule

Construct the full expression

Construct the derivative of g, first as two product-rule terms and then factored. Include the exponential chain factor and check the value at zero

1. 1Your written solution Write the complete rule, derivation and domain on paper. Written practice is ungraded

Task

\(\displaystyle g(x)=(2x-1)e^{3x}\)

Evaluate g′(0)

Review the whole expression

The two product-rule terms combine to give the factored derivative shown above. Both domains are R. A negative gradient at zero is consistent with the exact check

1. 1Review criteria Check every term and sign, the original domain and any excluded inputs, then independently substitute or differentiate to verify. Explain discrepancies and correct your own written expression

Construct the quotient in the correct order

Use denominator times derivative of numerator, minus numerator times derivative of denominator, over denominator squared. Retain the original denominator exclusion even after simplifying

Construct the full expression

Write and simplify the complete quotient-rule derivative of g. State its domain, check at zero, and explain the sign of the second numerator term before expanding

1. 1Your written solution Write the complete rule, derivation and domain on paper. Written practice is ungraded

Task

\(\displaystyle g(x)=\frac{2x+1}{x^2+1}\)

Evaluate g′(0)

Review the whole expression

The denominator is positive for every real input, so both domains are all real numbers. The numerator simplifies as shown above

1. 1Review criteria Check every term and sign, the original domain and any excluded inputs, then independently substitute or differentiate to verify. Explain discrepancies and correct your own written expression