(2) Find the derivatives of the given functions. 12 (a) h(w) = 3w? + w2 0- 1 (b) y% = (c) AP if V =ar2b dr 1. 4) 2) x2 - 4.x (d) H(x) = 10x +4 g3 + x2 +x- 2) (e) p(x) = (x² + 5x + 2)(x* - r + x? + x-2)

Part 1 1. Determine the slope of the following functions. i. f(x)=4x³+5x+-9x+2 ii. f(x)=sin(3x)-5x³-7 2. Let the velocity function of a moving object is V(t) = 7+3+ 5t² - 4t. What is the function for the acceleration of the object at time t.

15. Let a be a real number. We will explore the derivatives of the function f(x) = eax. By using the chain rule, we see (a) What is Recall that the second derivative of a function is the derivative of the deriva- tive function. Similarly, the third derivative is the derivative of the second derivative. (b) What is d² dx² d3 C030 BY NO SA d dx = aeax (eax), the third derivative of eax? (c) Let n be a natural number. Make a conjecture about the nth derivative dn of the function f (x) = eax. That is, what is (eax)? This con- dxn jecture should be written as a self-contained proposition including an appropriate quantifier. (eax), the second derivative of eax? 4.1. The Principle of Mathematical Induction (d) Use mathematical induction to prove your conjecture. 183

Derivative problems.