![]() ![]() This process will become clearer as you do the problems. (the term f'( g( x) ) ), then differentiate the inner layer (the term g'( x) ). Function f is the ``outer layer'' and function g is the ``inner layer.'' Thus, the chain rule tells us to first differentiate the outer layer, leaving the inner layer unchanged For example, it is sometimes easier to think of the functions f and g as ``layers'' of a problem. Instead, we invoke an intuitive approach. However, we rarely use this formal approach when applying the chain rule to specific problems. In the following discussion and solutions the derivative of a function h( x) will be denoted by or h'( x). The chain rule is a rule for differentiating compositions of functions. f 0 (x) (ln a)ex ln a Note: Problem 17 would help you to compute derivatives of the form ax, you just need to replace a in the last formula and you will get the derivative.The following problems require the use of the chain rule. Note: You can also use the chain rule, but this would lead to the same answer. ![]() (e4x )3 You can use the properties of the exponents and rewrite the function, e12x, then 0 (x) 12e12x. ![]() g(y) ln(y 3 ) ln(2y 2 ) going to use the property that ln(a b) ln a ln b, to rewrite the function. u(x) ) 0 Apply chain rule: u (x) 1 1 x 3 15. l(x) 2 x2 1 First rewrite as a power, then apply chain rule: l0 (x) ( 21 )(x2 (2x) 14. w(t) Practice sheet of 2 3t3 5t Apply quotient rule: w0 (t) (9t2 5)(t 9) (1)(3t3 5t) (t 9)2 x7 1 ln(x2 ) Again quotient rule, but as before use chain rule when calculating the derivative of (7圆 )(ln(x2 )) ( x12 )(2x)(x7 1) ln(x2 ). ![]() g(y) ey 2 Use the rule I display in class: g 0 (y) (2y)ey 2 8. f (t) (t2 8t) ln(3t) Apply product rule, remember we also chain when looking for the derivative of 1 ln(3t). w(x) (2x 4)(x 1)(7x 2) Multiply the first two parenthesis and apply product rule. h(x) (x 5)(x 8) We can use product rule or you can multiply and the use power rule, going to use the product rule. g(x) 5 x 1 First write 5th root as a power of x, g(x). f (x) 3x4 8 the Power Rule: f 0 (x) 12x3 7 5 2 x x Before we apply the power rule we rewrite the function: z(x) 3, then z 0 (x) 3. Preview text MAT 284 Spring 2017 Practice sheet Instructor: F Name: Compute the following derivatives power rule : f (x) xn f 0 (x) product rule : (f (x) g(x))0 f 0 (x) g(x) g 0 (x) f (x) Quotient rule : f (x) g(x) f 0 (x) g(x) g 0 (x) f (x) Chain rule : (f g 0 (x)f 0 (g(x)) 2 (g(x)) Solutions 1.
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