6.3.3 Logarithmic Differentiation Flashcards
Logarithmic Differentiation
- Understand that the Power Rule of Differentiation cannot be applied when the exponent is a variable.
- Take the natural log of both sides of an equation to simplify the problem by transforming a variable exponent into a product (of the variable with the natural log function).
- Apply logarithmic differentiation to complicated functions to make it easier to find derivatives by transforming exponents to products.
note
- The Power Rule of Differentiation, d/dx(x n ) = nx^n − 1 , may only be applied when the exponent, n, is a fixed number. If there is a variable in the exponent, as with the function y = x^x , the best approach is to turn the exponent into a product
by taking the natural log of both sides of the equation. - To find the derivative, differentiate the resulting equation implicitly with respect to x.
- The expression (1/y)·y ́ will always be on the left side of the equation. To solve for y ́, multiply both sides of the equation by y.
- Then on the right side of the equation, replace y with its equivalent expression in terms of the x that was given at the very beginning of the problem.
- Apply logarithmic differentiation whenever the function
whose derivative is sought has a variable appearing in an exponent. - In this example the steps are laid out carefully.
- First, take the natural log of both sides and pull out cos x in the exponent so that it becomes a term in the product.
- Second, implicitly differentiate both sides, remembering to apply the Product Rule of Differentiation and the Chain Rule as necessary.
- Third, multiply both sides by y and replace y with (sin x) cos x .
Differentiate the given function.
f(x)=x^x+1
f′(x)=x^x(xlnx+x+1)
Differentiate the given function.
f(x)=(2x+3)^5x−7
f′(x)=(2x+3)^5x−8 ((10x+15)ln(2x+3)+10x−14)
Differentiate the given function.
f(x)=(sinx)x^3x
f′(x)=(sinx)x^3x (cotx+3lnx+3)`
Differentiate the given function.
f(x)=x^tanx
f′(x)=x^(tanx)−1(x(sec2x)lnx+tanx)
Differentiate the given function.
f (x) = x^2x
f ′( x) = 2x^2x (ln x + 1)
Differentiate the given function.
f (x) = (cos x)^x
f ′( x) = (cos x)^x (ln (cos x) − x tan x)
Differentiate the given function.
f(x)=(cosx)^sinx
f′(x)=(cosx)^sinx ((cosx)ln(cosx)−sinxtanx)
Differentiate the given function.
f(x)=(sinx)^x
f′(x)=(sinx)^x (ln(sinx)+xcotx)
Differentiate the given function.
f(x)=(√7x−1)x^2
f′(x)=(√7x−1)^x^2 ((14x^2−2x)ln(7x−1)+7x^2/14x−2)
Differentiate the given function.
f(x)=x(tanx)^x
f′(x)=(tanx)^x(1+xln(tanx)+x^2cscxsecx)
Differentiate the given function.
f(x)=x^√2x+1
f′(x)=x^√2x+1 (xlnx+2x+1/x√2x+1)
Differentiate the given function.
f(x)=(lnx)^lnx
f′(x)=(lnx)^lnx/x (ln(lnx)+1)
