Derivatives!

1. Derivatives! By: Emily Lael Halsmer

2. What is a Derivative Review • Specific things a derivative will tell you • The slope of the tangent line to a curve • The rate that something is changing Definition: The derivative of a function f at a number a, denoted f’(a), is the limit, as x approaches infinity, of f(x)-F(a) divided by x-a (if the limit exists) Or you may use this more spacifice

3. Another way to look at the Definition • Or you may use this more specific definition • F’(a) is the limit as h approaches zero of f(a+h)-f(a) all divided by h • Remember the word slope is always equal to the first derivative of a function • This concept explained is that the first derivative of a function, F’(x), at a certain point (x,y) is the slope of the tangent line at (x,y) on the curve F(x).

4. Taking Derivative Rules! (shortcuts) • Product Rule • Definition: If g and f are both differentiable, then d/dx[f(x)g(x)] = f(x) d/dx[g(x)] + g(x) d/dx[f(x)] This looks all complicated but don’t worry and remember this little saying… “first times the derivative of the second, plus the second times the derivative of the first” Example: g(x) = (x^2)(e^x) g’(x) = (x^2)(e^x) + (e^x)(2x)

5. Now you try on your own • Find the first derivative of f(x)= [sqrt(x)][e^x] • F’(x)= [sqrt(x)][(1/2)e^x] + [e^x][(x^(-1/2)] • F’(x)= [sqrt(x)][e^x] + [e^x][(1/2) x^(-1/2)] • F’(x)= [sqrt(x)][e^x] + [e^x][(3/2) x^(-1/2)] • F’(x)= [sqrt(x)][e^x] + [e^x][(1/2) x^(1/2)] *You may use scratch paper to work out the problem, if needed

6. Correct!!!! • Congratulations! You are on your way to understanding the steps to use the Product Rule! • Your answer: F’(x)= [sqrt(x)][e^x] + [e^x][(1/2) x^(-1/2)]

7. Incorrect • You picked the wrong answer. • The correct answer is F’(x)= [sqrt(x)][e^x] + [e^x][(1/2) x^(-1/2)]

8. Question 2: more practice • Find the Derivative, using the product rule of… F(x)= e^(3x^2) x^4 • F’(x)= [x^4][(6x) e^(3x^2)] + [e^(3x^x)][4x^3] • F’(x)= [e^(3x^x)][4x^3] - [x^4][(6x) e^(3x^2)] • F’(x)= [e^(3x^x)][x^4] +[(6x) e^(3x^2)][4x^3] • F’(x)= [e^(3x^x)][4x^3] + [x^4][(6x) e^(3x^2)] Hint: You will also need to do the chain rule

9. Correct!! • Congratulations! • You have chosen F’(x)= [e^(3x^x)][4x^3] + [x^4][(6x) e^(3x^2)] • Here are the solution steps in case you want to review. • Solution Steps: Break F(x) into 2 sections • F(x)= [e^(3x^2)] [x^4] • Now use the product rule: if g(x)= e^(3x^2) g’(x)= (6x) e^(3x^2) And if h(x)= x^4 h’(x)= 4 x^3 • Using this information you are able to say F’(x)= g(x) h’(x) + h(x) g’(x) • And get the correct solution

10. Incorrect • This problem is more complicated than the first because you are also using the chain rule along with the product rule. • The answer is F’(x)= [e^(3x^x)][4x^3] + [x^4][(6x) e^(3x^2)] • Solution Steps: Break F(x) into 2 sections • F(x)= [e^(3x^2)] [x^4] • Now use the product rule: if g(x)= e^(3x^2) g’(x)= (6x) e^(3x^2) And if h(x)= x^4 h’(x)= 4 x^3 • Using this information you are able to say F’(x)= g(x) h’(x) + h(x) g’(x) • And get the correct solution

11. You are finished! • Hopefully now you have a better understanding of how things work using the product rule to find the derivatives of functions. • Remember that when you plug in a point (x,y), to the first derivative, F’(x), that gives you the slope at that point (x,y), on the curve F(x)