3.6
Derivatives of logarithmic functions
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3.6 (#31)
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• (ln(|x|))’= 1/x
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• Calculate the derivatives of the following
functions:
• cos(ln(x))
• ln(cos(x))
• ln(ln(ln(x))
• ln((x+1)/(x-1))
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Logarithmic differentiation
Find the derivatives of the following functions
• x^x
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To find the derivative of f(x) using logarithmic
differentiation (useful if f(x) has a formula of the
type g(x)^h(x):
• Write y = f(x)
• Take logs of both sides :
ln(y)=ln(f(x))= h(x) ln(g(x))
• Differentiate implicitly with respect to x :
y’/y = h’(x)ln(g(x))+h(x)g’(x)/g(x)
• Solve for y’ (replacing y with f(x))
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Calculate the derivative of the function
(x+1)^(cos(x)*sin(x))
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4. [10 points] Find the equation for the tangent line to the following curve at the point (3, 0):
3ey + x2 y = x
5. [5 points] Consider the curve f pictured below:
f (x)
5
4
3
2
1
-4
-3
-2
-1
0
1
2
3
4
5
6
7
x
-1
Let h(x) = ln(f (x)). Compute h0 (4).
(An estimate is okay. Show your work by explaining your reasoning, or by clearly marking the
relevant information on the graph.)
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