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Wize University Calculus 1 Textbook > Derivatives

Logarithmic Differentiation

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Logarithmic Differentiation

Consider a function of the form y=g(x)f(x).y=g(x)^{f(x)}. To compute the derivative of this function, it is useful (and sometimes necessary) to first take the logarithm on both sides of the equation and then compute the derivative. This process is called logarithmic differentiation.

Procedure for Logarithmic Differentiation

  1. Take the natural logarithm on both sides of the equation: ln⁡y=ln⁡ (g(x)f(x))\ln y=\ln\ (g(x)^{f(x)})
  2. Simplify using log rules
  3. Differentiate Implicitly
  4. Solve
  5. Simplify
  6. Substitute given information (if asked to)
Wize Tip
You can use Logarithmic Differentiation on derivatives of complex products or quotients by using properties of logarithms.

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Example: Logarithmic Differentiation

Find the derivative of f(x)=xxf(x)=x^x.


1) ln⁡f(x)=ln⁡xx\text{1)} \ \ln f(x) = \ln x^x

2) ln⁡f(x)=xln⁡x\text{2)} \ \ln f(x) = x \ln x
3) f′(x)f(x)=1×ln⁡x+x×1x\text{3)} \ \displaystyle \frac{f'(x)}{f(x)}=1\times\ln x + x \times\frac{1}{x}

4) f′(x)=f(x)[ln⁡x+1]\text{4)} \ f'(x)=f(x)[\ln x +1]

5) f′(x)=xx[ln⁡x+1]\text{5)} \ f'(x)=x^x[\ln x +1]

Find the derivative of f(x)=sin⁡xcos⁡xf(x)=\sin{x}^{\cos{x}}.

Extra Practice
Derivatives
Find the derivative of the following functions at the given values
a) f(x)=arcsin⁡(x2)x3−1f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1} at x=0x=0
b) g(x)=(cos⁡x)exg\left(x\right)=\left(\cos x\right)^{e^x} at x=0x=0
Logarithmic Differentiation
If f(x)=xxf(x)=x^{\sqrt{x}} then find f′(4)f'(4).
Implicit Differentiation: Logarithmic Differentiation
Find the equation of the tangent line to
(sin⁡(xy))x=x1/4(\sin(xy))^x = x^{1/4}
at the point (12,π2)\left(\dfrac{1}{2},\dfrac{\pi}{2}\right).
Logarithmic Differentiation
Use logarithmic differentiation to find g′g' where g(x)=xx22+tan⁡xg(x)=x^{x^2}\sqrt{2+\tan x}
Logarithmic Differentiation
If f(x)=(2x)sin⁡xf(x)=(2x)^{\sin x} , find f′(π2)f'(\frac{\pi}{2}) ?
Logarithmic Differentiation
If f(x)=xxf(x)=x^{\sqrt{x}} then find f′(4)f'(4).
Logarithmic Differentiation
Use logarithmic differentiation to find g′g' where g(x)=xx22+tan⁡xg(x)=x^{x^2}\sqrt{2+\tan x}
Complicated Log Derivative
If f(x)=(sin⁡x)3xf\left(x\right)=\left(\sin x\right)^{3x}, then f′(x)=f'\left(x\right)=
Logarithmic Differentiation: Big Product/Quotient
Consider y=(x2+1)2(x−1)4(x3+2)33\displaystyle y=\sqrt[3]{\frac{(x^{2}+1)^{2}(x-1)^{4}}{(x^{3}+2)^{3}}}. Compute dydx at x=0\displaystyle\frac{dy}{dx}\text{ at }x=0.
Logarithmic Differentiation
Find the derivative of y=(x+1)(x−1)y=\left(x+1\right)^{\left(x-1\right)}
Logarithmic differentiation
Find the derivative of the function
y=sin⁡(x)sin⁡(x)y = \sin(x)^{\sin(x)}
Logarithmic Differentiation
Find the derivative of y=(x+1)(x−1)y=\left(x+1\right)^{\left(x-1\right)}
Complicated Log Derivative
If f(x)=(sin⁡x)3xf\left(x\right)=\left(\sin x\right)^{3x}, then f′(x)=f'\left(x\right)=
Logarithmic Differentiation
Find the derivative of y=(x+1)(x−1)y=\left(x+1\right)^{\left(x-1\right)}
Logarithmic Differentiation
Find the derivative of y=(x+1)(x−1)y=\left(x+1\right)^{\left(x-1\right)}
Logarithmic differentiation
Find the derivative of the function
y=sin⁡(x)sin⁡(x)y = \sin(x)^{\sin(x)}
Logarithmic Differentiation
Find the derivative of y=(x+1)(x−1)y=\left(x+1\right)^{\left(x-1\right)}
Complicated Log Derivative
If f(x)=(sin⁡x)3xf\left(x\right)=\left(\sin x\right)^{3x}, then f′(x)=f'\left(x\right)=
Complicated Log Derivative
If f(x)=(sin⁡x)3xf\left(x\right)=\left(\sin x\right)^{3x}, then f′(x)=f'\left(x\right)=
Complicated Log Derivative
If f(x)=(sin⁡x)3xf\left(x\right)=\left(\sin x\right)^{3x}, then f′(x)=f'\left(x\right)=
Logarithmic Differentiation
Find the derivative of y=(x+1)(x−1)y=\left(x+1\right)^{\left(x-1\right)}
Logarithmic differentiation
Find the derivative of the function
y=sin⁡(x)sin⁡(x)y = \sin(x)^{\sin(x)}
Practice: Log Differentiation
Practice: Log Differentiation
Find the derivative of x2(cos⁡x)(e2x)2xln⁡(x)\frac{x^2\left(\cos x\right)\left(e^{2x}\right)}{2x\ln\left(x\right)}
Complicated Log Derivative
If f(x)=(sin⁡x)3xf\left(x\right)=\left(\sin x\right)^{3x}, then f′(x)=f'\left(x\right)=
Logarithmic Differentiation
Use logarithmic differentiation to find g′g' where g(x)=xx22+tan⁡xg(x)=x^{x^2}\sqrt{2+\tan x}
Logarithmic Differentiation
Find y′y' for the function 𝑦=𝑥2−x2𝑦 = 𝑥^{2-x^2}.
Find the derivative of y=(cos⁡x)5x2y=\left(\cos x\right)^{\frac{5x}{2}}
Logarithmic for Implicitly Defined Function
Find dy/dxdy/dx at the point (1,1)(1,1) given xy=yxx^y=y^x
Logarithmic Differentiation: Big Product/Quotient
Consider y=(x2+1)2(x−1)4(x3+2)33\displaystyle y=\sqrt[3]{\frac{(x^{2}+1)^{2}(x-1)^{4}}{(x^{3}+2)^{3}}}. Compute dydx at x=0\displaystyle\frac{dy}{dx}\text{ at }x=0.
Logarithmic with Chain
Use logarithmic differentiation to find g′g^\prime where g(x)=xx2+sin⁡x\displaystyle g(x)=x^x\sqrt{2+\sin{x}}.
Differentiation: Logarithmic with Trig
If y=sin⁡xxy=\sin^xx , find y′y'.
Derivatives: Logarithmic Functions
Compute the derivative of f(x)=xx+1f(x) = x^{x + 1}. Remember that log⁡x=log⁡ex=ln⁡x\log x = \log_e x = \ln x.
Derivatives
Find the derivative of the following functions at the given values
a) f(x)=arcsin⁡(x2)x3−1f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1} at x=0x=0
b) g(x)=(cos⁡x)exg\left(x\right)=\left(\cos x\right)^{e^x} at x=0x=0
Logarithmic Differentiation
Find the derivative of y=(x+1)(x−1)y=\left(x+1\right)^{\left(x-1\right)}
Implicit Differentiation: Logarithmic Functions
For the following equations, solve for dydx\frac{dy}{dx} :
a) y=(5x2+3)sin⁡(x)y = (5x^2 + 3)^{\sin(x)}
b) x2y+3y2x2=x+yx^2y + 3y^2 x^2 = x + y
Logarithmic Differentiation
Calculate the derivative of the following functions.
f(x)=(tan⁡x)secx\displaystyle f(x)=(\tan{x})^{\text{sec}{x}}
g(x)=(xtan⁡4x)xg(x)=(x\tan{4x)}^x
Implicit Differentiation: Logarithmic Differentiation
Find the equation of the tangent line to
(sin⁡(xy))x=x1/4(\sin(xy))^x = x^{1/4}
at the point (12,π2)\left(\dfrac{1}{2},\dfrac{\pi}{2}\right).
Logarithmic Differentiation
Find the derivative of
f(x)=(tan⁡x)ln⁡xf(x)=(\tan x)^{\ln x}
Logarithmic Differentiation
Use logarithmic differentiation to find g' where
g(x)=xx2+sin⁡xg(x)=x^x\sqrt{2+\sin x}
Logarithmic Differentiation
Compute the derivative of
f(x)=x3/2x+1(x+5)7sin⁡xf(x)=\frac{x^{3/2}\sqrt{x+1}}{(x+5)^7\sin x}
Find the slope of the tangent line to g(x)=xx2+sin⁡xg(x)= x^x\sqrt{2+\sin x} when x=πx=\pi .
Differentiate the following functions.
(a) g(t)=(t5+t2)6tg(t)=(t^5+t^2)^{6t} (b) f(x)=x6xf(x)=\sqrt x^{6x} (c) f(x)=(x3+4)tan⁡xf(x)=(x^3+4)^{\tan x} (d) Finddydx\frac{dy}{dx} if ycos⁡(x)=xtan⁡(y)y\cos(x)=x\tan(y)
Calculate the derivative of the following function.
f(x)=(sin⁡x)cos⁡x\displaystyle f(x)=(\sin{x})^{\cos{x}}
Logarithmic Differentiation
Find y′y' for the function 𝑦=𝑥2−x2𝑦 = 𝑥^{2-x^2}.
Logarithmic Differentiation
Use logarithmic differentiation to find g′g' where g(x)=xx22+tan⁡xg(x)=x^{x^2}\sqrt{2+\tan x}
Compute the derivative of f(x)=x3/2x+1(x+5)7sin⁡xf(x) = \dfrac{x^{3/2}\sqrt{x+1}}{(x+5)^7\sin x}
Find the derivative of y=(cos⁡x)5x2y=\left(\cos x\right)^{\frac{5x}{2}}
Logarithmic Differentiation: Big Product/Quotient
Consider y=(x2+1)2(x−1)4(x3+2)33\displaystyle y=\sqrt[3]{\frac{(x^{2}+1)^{2}(x-1)^{4}}{(x^{3}+2)^{3}}}. Compute dydx at x=0\displaystyle\frac{dy}{dx}\text{ at }x=0.
Logarithmic for Implicitly Defined Function
Find dy/dxdy/dx at the point (1,1)(1,1) given xy=yxx^y=y^x
Logarithmic with Chain
Use logarithmic differentiation to find g′g^\prime where g(x)=xx2+sin⁡x\displaystyle g(x)=x^x\sqrt{2+\sin{x}}.
Differentiation: Logarithmic with Trig
If y=sin⁡xxy=\sin^xx , find y′y'.
Logarithmic Differentiation
Consider y=(x2+1)2(x−1)4(x3+2)33\displaystyle y=\sqrt[3]{\frac{(x^{2}+1)^{2}(x-1)^{4}}{(x^{3}+2)^{3}}}. Compute dydx at x=0\displaystyle\frac{dy}{dx}\text{ at }x=0.
Logarithmic differentiation
Find the derivative of f(x)=(x+1)9ex(sin⁡ x)32x5x2+1f\left(x\right)=\frac{\left(x+1\right)^9e^x\left(\sin\ x\right)^{\frac{3}{2}}}{x^5\sqrt{x^2+1}}.
Logarithmic Differentiation
Find the derivative of y=(x−3)9 xx ln⁡ xx−9.y=\frac{\left(x-3\right)^9\ x^x\ \ln\ x}{x-9}.
Logarithmic Differentiation
If f(x)=(2x)sin⁡xf(x)=(2x)^{\sin x} , find f′(π2)f'(\frac{\pi}{2}) ?
Logarithmic Differentiation
If f(x)=xxf(x)=x^{\sqrt{x}} then find f′(4)f'(4).
Find the derivative of f(x)=(tan⁡x)ln⁡ x.f(x)=(\tan x)^{\ln\ x}.
Find the derivative of f(x)=(tan⁡x)ln⁡xf(x)=(\tan x)^{\ln x}.
Calculate the derivative of the following functions.
f(x)=(tan⁡x)secx\displaystyle f(x)=(\tan{x})^{\text{sec}{x}}
Find f′(1)f'(1) if f(x)=(arctan⁡x)x2f\left(x\right)=\left(\arctan x\right)^{x^2}.
Practice: Log Differentiation
Practice: Log Differentiation
Find the derivative of x2(cos⁡x)(e2x)2xln⁡(x)\frac{x^2\left(\cos x\right)\left(e^{2x}\right)}{2x\ln\left(x\right)}
Complicated Log Derivative
If f(x)=(sin⁡x)3xf\left(x\right)=\left(\sin x\right)^{3x}, then f′(x)=f'\left(x\right)=
Logarithmic differentiation
Find the derivative of the function
y=sin⁡(x)sin⁡(x)y = \sin(x)^{\sin(x)}
Logarithmic Differentiation
Calculate the derivative of the following functions.
f(x)=(sin⁡x)cos⁡x\displaystyle f(x)=(\sin{x})^{\cos{x}}
Logarithmic Differentiation
Calculate the derivative of the following functions.
f(x)=xx2+sin⁡x\displaystyle f(x)=x^x\sqrt{2+\sin{x}}
Logarithmic Differentiation
Calculate the derivative of the following functions.
f(x)=(tan⁡x)secx\displaystyle f(x)=(\tan{x})^{\text{sec}{x}}
Practice: Logarithmic Differentiation
Find the derivative of f(x)=sin⁡xcos⁡xf(x)=\sin{x}^{\cos{x}}