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Section7_4_review

# Section7_4_review - Section 7.4 Derivatives of Logarithmic...

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Section 7.4: Derivatives of Logarithmic Functions Differentiation/Integration Rules obtained in this section: d dx ln x ( 29 = 1 x d dx ln u ( 29 = 1 u du dx , where u is a function of x d dx log a x ( 29 = 1 x ln a d dx log a u ( 29 = 1 u ln a du dx , where u is a function of x d dx a x ( 29 = a x ln a d dx a u (29 = a u ln a du dx , where u is a function of x 1 x dx = ln x + c tan xdx = lnsec x + c a x dx = a x ln a + c Logarithmic Differentiation (Used to differentiate functions of the form ‘a function of x raised to a function of x ’ and complicated functions that involve multiple products, quotients, or powers.) (i) Write the function as y = f x (29 . (ii) Take the natural log of both sides, and use the properties of logs to expand. (iii) Differentiate implicitly with respect to x, keeping in mind that d dx ln y ( 29 = 1 y dy dx . (iv) Solve for d y d x . Problem. Differentiate y = 10 tan θ . Solution. Since d dx a u = a u ln a du dx , y = 10 tan θ ln10 d d θ tan θ ( 29 = 10 tan θ ln10 ( 29 sec 2 θ Problem. Differentiate y = ln x 4 sin 2 x ( 29 .

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