Derivative Anti-derivativessin(x) -cos(x)cos(x) sin(x)sec2(x) tan(x)csc2(x) -cot(x)sec(x)tan(x) sec(x)csc(x)cot(x) -csc(x) Xn+1 Xn n+1 f’(g(x))g’(x) f(g(x)) f’(x) is also the same as the integral symbol ∫
Derive: f(x)=3x2+2x Answer: f’(x)= 6x+2 Now Anti-Derive: ∫6x+2dx Answer: f(x)=3x2+2x+c Where did the “c” come from? The “c” means any constant. When you derive a constant in a function, the derivative is 0,so when anti-deriving always add “+c” at the end because you cannot assume whether or not there was a constant in the original function, and by adding “c” you are making sure you didn’t leave any numbers out of the function. RULES
Anti-derive the following: ∫sin(x)dx -cos(x) + c ∫csc2(X)dx -cot(x) + c ∫sec(x)tan(x)dx sec(x) + c ∫4x + cos(x)dx 2x2 + sin(x) + c Is there a rule for “4x” (one something to a power)? YES! If you don’t remember click the button 4x= 4x2 = 2x2 2 RULES
∫3x2 - 7x + 4 - 5sec2(x)dx Step by Step If you forgot your rules… Rule for 3x2? Yes x3 Rule for 7x? Yes 7x2 2 Rule for 4? Yes 4x Rule for 5sec2(x)? Yes 5tan(x) Put it all together… X3 - 7x2 + 4x -5tan(x) + c 2 RULES
If the function u=g(x) has a continuous derivative on the closed interval [a,b] and f is continuous on the range of g, then ∫ab f(g(x))g’(x)dx = ∫g(a)g(b) f(u)du Anti-derive: ∫(x2+4) 9 (2x)dx Is there are rule for this one? Of course, it’s the Product Rule…WRONG! If this is what you were thinking then Click! You have to use u-substitution to solve this problem. Derivative u= x2+4 and du=2xdx Now Substitute!
u= x2+4 du=2xdx ∫(x2+4) 9 (2x)dx ∫(u) 9du This is your new equation, now is there a rule for this? Yes, so Anti-derive. u 10 + c Not finished yet…now plug back in your original numbers. 10 (x2+4)10 + c This is the final answer 10 RULES
∫5cos(5x)dx u=5x du=5dx ∫cos(u)du sin(u) + c sin(5x) + c ∫x(5x 2 - 3)7dx u=5x 2 – 3 du=10xdx but there is not 10x in the problem?!? That’s ok, 10 is a coefficient so just move it! 1/10du=xdx Now that we have everything…rewrite the problem ∫u7 (1/10)du (1/10)u8 u8 + c 8 80 Don’t forget to plug the original numbers back in … (5x 2 - 3)8 + c 80 RULES