3.4 The Chain Rule. F (x) = f (g(x))g (x) Alternate way of thinking about it: If y = f(u) and u = g(x) where both are differentiable functions, then

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1 3.4 The Chain Rule To find the derivative of a function that is the composition of two functions for which we already know the derivatives, we can use the Chain Rule. The Chain Rule: Suppose F (x) = f(g(x)). Then, provided that g (x) and f (g(x)) both exist, F (x) = f (g(x))g (x) Alternate way of thinking about it: If y = f(u) and u = g(x) where both are differentiable functions, then Find the derivatives of the following functions. dy dx = dy du du dx f(x) = x x 3 + x f(x) = ( ) 5x x 3 x f(x) = tan 5x sec 3x h(x) = cos(x 3 ) + cos 2 x + sin 4 (2x) h(x) = sec(sin 2x) f(x) = tan 3 (cot 4x) 1

2 g(x) = e 3x + xe x + e x + cos(e x2 ) Find the first and second derivatives of h(x) = e tan x. One new rule: d dx (ax ) = a x ln a Find the derivative of f(x) = 3 x + 4 x5 +8 Suppose that F (x) = f(g(x)) where f (4) = 3, f (5) = 2, g (4) = 6 and g(4) = 5. Calculate F (4). [ ] 2 Find a formula for G (x) if G(x) = f(sin x) + g(e x/2 ) 2

3 Find the 37th derivative of f(x) = sin 4x. Find the 100th derivative of f(x) = xe x Find f (15) (x) for f(x) = 1 3x + 1 3

4 3.5 Implicit Differentiation and Derivatives of Inverse Trig Functions When y cannot be written explicitly as a function of x (or not easily), we can use the method of implicit differentiation. To find dy dx, differentiate both sides with respect to x, remembering that the Chain Rule is necessary since y is dependent on x. Find dy dx for the equation x2 + y + y 4 = 16. Find y for the equation (y 2 + 1) 3 + xy = 3x 2 2y. What is the slope of the tangent line at the point (2, 1)? Find dy dx for the equation sin 2x 5x4 y 2 = cos 3y 4

5 Find dy dx for the equation ey + y cos x = e sin y. Example: Find an equation of the tangent line to the hyperbola y2 36 x2 4 = 1 at the point (1, 3 5). Derivatives of Inverse Trig Functions d dx (sin 1 x) = 1 1 x 2 d 1 dx (cos 1 x) = 1 x 2 d dx (tan 1 x) = x 2 Show that d dx (sin 1 x) = 1 1 x 2. 5

6 Find the derivatives of the following functions. f(x) = arcsin(5x 2 + 1) h(x) = arccos( x) k(x) = x arctan(e x ) Find the equation of the tangent line to the graph of f(x) = arctan(2x) at the point where x =

7 3.6 Derivatives of Logarithmic Functions d dx (ln x) = 1 x d dx (ln x ) = 1 x Chain Rule Version: d dx log a x = 1 ln a 1 x = 1 x ln a Chain Rule Version: d dx ln(g(x)) = g (x) g(x) d dx log a(g(x)) = g (x) g(x) ln a Find the equation of the tangent line to the graph of f(x) = ln(3x 2 2 x) at the point where x = 1. Find the derivatives of the following functions. g(x) = log 2 (x 4 5x) h(x) = x ln(cos x) f(x) = ln(ln 4x) h(x) = log((x 3 + 7x)(e x + x 7 )) 7

8 g(t) = ln ( t 2 4 ) (t 3 7t) 5 Logarithmic Differentiation: Sometimes it is easier to differentiate a function by first taking the logarithm of both sides, differentiating implicitly and then solving for y. Use this method when: (1) The function is a quotient or product of a lot of terms. Log. Diff. recommended, but not necessary. (2) The function is of the form y = f(x) g(x). Logarithmic Differentiation NECESSARY. Find the derivatives of the following functions: x f(x) = (2x 1) 4 tan 7 x f(x) = x cos x f(x) = (sin x) ex 8

9 Vector Supplement II Part 1: Derivatives of Vector Functions If r(t) = x(t), y(t) is a vector function, then r (t) = x (t), y (t) if both of these derivatives exist. r (a) represents a tangent vector to the curve where t = a. The tangent vector also points in the direction the curve is traced out as t increases. Find the domain of the vector function r(t) = t t 6, 3t 5. Given the vector function r(t) = t 2 + t, 4t 2, find a tangent vector to the curve at the point where t = 3. Find parametric equations for the tangent line to the curve where t = 3. 9

10 Example: Find a unit tangent vector to the curve r(t) = t sin t, 4 2 cos 3t at the point where t = π 2. Find a unit tangent vector to the curve r(t) = 5t 2 + 1, 8t 2 t at the point (6, 9). If r(t) = x(t), y(t) is a vector function representing the position of a particle at time t, then the velocity is given by v(t) = r (t) and the acceleration is given by a(t) = r (t). The speed of the particle is the magnitude of the velocity: r (t). Example: A projectile is fired so that its position is given by the function r(t) = e 2t, 4t 2 t. Find the velocity, speed, and acceleration of the projectile at time t. 10

11 Vector Supplement II Part 2: Slopes and Tangents to Parametric Curves Consider r(t) = x(t), y(t) and its derivative at t = a, r (a) = x (a), y (a). This vector is a tangent vector to the graph at the point where t = a. What would be the slope of the tangent line? So, without initially eliminating the parameter, we can find the slope of the tangent line, dy, by using dx dy dx = y (t) x (t) = dy/dt dx/dt Example: Find dy dx for the curve defined parametrically by x = ln(t2 + 1), y = e t cos t. Example: Find an equation of the tangent line to the curve r(t) = t 2 + t, t when t = 4. 11

12 Example: What is the slope of the tangent line to the curve x = 3t 2 8, y = 5t 2 + 2t at the point (4, 16). Find the points on the curve where the tangent line has a slope of 2. A curve has a horizontal tangent where: A curve has a vertical tangent where: Example: Find all points on the curve defined by the parametric equations x = t 3 12t and y = 2t 3 9t 2 where the tangent line is horizontal or vertical. 12

13 3.7 Rates of Change in the Natural and Social Sciences Consider the graph of a position function s(t) for a particle that starts off moving horizontally forward. When is the particle moving forward? s When is the particle moving backward? 4 2 When is the particle at rest? t An object is thrown vertically upward with a velocity of 40 ft/s. s(t) = 40t 16t Its height after t seconds is given by What is the maximum height achieved by the object? What is the object s velocity when it is at a height of 40 ft on its way down? With what velocity does the object hit the ground? 13

14 A particle moves according to the position function s(t) = 2t 3 12t t + 3, for t 0 where t is measured in seconds and s is measured in feet. When is the particle at rest? When is the particle moving in the positive direction? negative direction? Find the total distance traveled in the first 5 seconds. A spherical balloon is being blown up. Find the rate at which the volume of the sphere is changing with respect to the radius when the radius of the balloon is 2 in. 14