AI HL · 13.5
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Anti-differentiation: reversing the power rule
13.5.2
Find f(x) when (a) \(f'(x)=x^{6}\), (b) \(f'(x)=x^{-5}\).
Have a go first. Then open the working one line at a time.
\(f(x)=\frac{x^{7}}{7}+c\)
Add 1 to the power: 6 + 1 = 7. Divide by the new power, 7.
\(-5+1=-4\)
Adding 1 to a negative power moves it towards zero.
\(1\div(-4)=-\frac14\)
Divide by the new power, which is negative.
\(f(x)=-\frac14x^{-4}+c\)
Do not forget c.
Find (a) f(x) when \(f'(x)=0\), (b) f(x) when \(f'(x)=1\), (c) y when \(\frac{dy}{dx}=7\).
Have a go first. Then open the working one line at a time.
\(f(x)=c\)
Only a constant has derivative 0. Every constant works, so the answer is c.
\(f(x)=x+c\)
Think of 1 as \(x^{0}\): add 1 to the power, divide by 1.
\(y=7x+c\)
A constant k integrates to kx.
Find y when \(\frac{dy}{dx}=\frac32x^{5}\).
Have a go first. Then open the working one line at a time.
\(5+1=6\)
The power goes up by 1.
\(\frac32\div6=\frac14\)
Dividing by 6 is multiplying by \(\frac16\): \(\frac32\times\frac16=\frac14\).
\(y=\frac14x^{6}+c\)
A coefficient that is a fraction stays a fraction.
Find \(\int \frac35y^{-4}\,dy\).
Have a go first. Then open the working one line at a time.
\(-4+1=-3\)
The variable is y but the rule is the same.
\(\frac35\div(-3)=-\frac15\)
A positive number divided by a negative one is negative.
\(-\frac15y^{-3}+c\)
dy tells you the variable is y.
If \(\frac{dy}{dx}=6x^{5}+8x^{3}+5\), find an expression for y.
Have a go first. Then open the working one line at a time.
\(\frac{6}{6}x^{6}+\frac{8}{4}x^{4}+\frac{5}{1}x^{1}\)
Treat 5 as \(5x^{0}\): the power becomes 1 and you divide by 1.
\(x^{6}+2x^{4}+5x\)
A coefficient of 1 is not written.
\(y=x^{6}+2x^{4}+5x+c\)
One constant for the whole answer.
Find \(\int(2x^{-5}+3x)\,dx\).
Have a go first. Then open the working one line at a time.
\(\frac{2}{-4}x^{-4}=-\frac12x^{-4}\)
Power −5 becomes −4.
\(\frac{3}{2}x^{2}\)
Power 1 becomes 2.
\(-\frac12x^{-4}+\frac32x^{2}+c\)
Each term is done on its own.
Find f(x) when \(f'(x)=-\frac{6}{x^{4}}\).
Have a go first. Then open the working one line at a time.
\(f'(x)=-6x^{-4}\)
\(\frac1{x^{4}}=x^{-4}\). You can only use the rule on a power.
\(\frac{-6}{-3}x^{-3}=2x^{-3}\)
New power \(-4+1=-3\).
\(f(x)=2x^{-3}+c\)
You may also write \(\frac{2}{x^{3}}+c\).
Find f(x) when \(f'(x)=6\sqrt{x}+4x^{-1/2}\).
Have a go first. Then open the working one line at a time.
\(f'(x)=6x^{1/2}+4x^{-1/2}\)
\(\sqrt{x}=x^{1/2}\).
\(\frac{6}{3/2}x^{3/2}=4x^{3/2}\)
\(\frac12+1=\frac32\). Dividing by \(\frac32\) is multiplying by \(\frac23\).
\(\frac{4}{1/2}x^{1/2}=8x^{1/2}\)
\(-\frac12+1=\frac12\).
\(f(x)=4x^{3/2}+8x^{1/2}+c\)
The same rule works for any power except −1.