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More on Derivatives and Integrals -Product Rule -Chain Rule AP Physics C Mrs. Coyle
f ’ (x) = lim f( x + h) f( x ) h 0 h Derivative
Derivative Notations f ’ (x) df (x) dx. f df dx
Notations when evaluating the derivative at x=a f(a) df (a) dx f’(a) df | x=a dx
Basic Derivatives d(c) = 0 dx d(mx+b) = m dx d(x n ) = n x n-1 dx n is any integer x≠0
Derivative of a polynomial. For y(x) = ax n dy = a n x n-1 dx - Apply to each term of the polynomial. -Note that the derivative of constant is 0.
Product Rule For two functions of x: u(x) and v (x) d [u(x) v (x)] =u d v (x) + v d u (x) dx dx dx or (uv)’ = u v’ + vu’
Example of Product Rule: Differentiate: F=(3x-2)(x 2 + 5x + 1) Answer: F’(x) = 9x x-7
Chain Rule If y=f(u) and u=g(x): dy = dy du dx du dx
Example of Chain Rule Differentiate: F(x)= (x 2 + 1) 3 Ans:F’(x)= 6(x 2 +1) 2 x
Second Derivative Notations df ’ (x) dx d 2 f (x) d x 2 f’’(x)
Example of Second Derivative Compute the second derivative of y=(x) 1/2 Ans: (-1/4) x -3/2
Derivatives of Trig Functions d sinx = cosx dx d cosx = -sinx dx d tanx = sec 2 x dx d secx = secx tanx dx
Derivative of the Exponential Function d e u = e u du dx dx
Example of derivative of Exponential Function 2 Differentiate: e x 2 Ans: 2x e x
Derivative of Ln d (lnx) = 1/x dx
Definite Integral b a ∫ b f(x) dx= F(b)-F(a)= F(x)| a a and b are the limits of integration.
If F(x)= ∫ f(x) dx then d F(x) = f(x) dx
Properties of Integrals a ∫ c f(x) dx = a ∫ b f(x) dx+ b ∫ c f(x) dx a
Basic Integrals (integration constant ommited) ∫ x n dx = 1 x n+1, n ≠ 1 n+1 ∫ e x dx = e x ∫ (1/x) dx = ln|x| ∫ cosx dx = sinx ∫ (1/x) dx = ln|x| ∫ sinx dx = -cosx
Example with computing work. There is a force of 5x 2 –x +2 N pulling on an object. Compute the work done in moving it from x=1m to x=4m. Ans: 103.5N
To evaluate integrals of products of functions : Chain Rule Integration by parts Change of Variable Formula
When a function and its derivative appear in the integral: a ∫ b f[g(x)]g’(x) dx = g(a) ∫ g(b) f(y) dy
Example: When a function and its derivative appear in the integral: Compute x=0 ∫ x=1 2x (x 2 +1) 3 dx Ans: 3.75 Ans:
Example of Change of Variable Formula Evaluate: 0 ∫ 1 2x (x 2 + 1) 9 dx Answ: 102.3
Integration by Parts a ∫ b u(x) dv dx= dx b = u(x) v(x)| a - a ∫ b v(x) du dx dx
Integration by Parts b a ∫ b u v’ dx= u v| a - a ∫ b v u’ dx
Example of Integration by Parts Compute 0 ∫ π x sinx dx Ans: π
Differentiation of the Exponential Function (e x ) and Natural Logarithms (lnx) Exponential function e x.
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Warm up Problem If, find.. The Derivative Function Def. For any function f (x), we can find the derivative function f (x) by:
2.4 The Chain Rule If f and g are both differentiable and F is the composite function defined by F(x)=f(g(x)), then F is differentiable and F is given.
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Section 3.3a. The Do Now Find the derivative of Does this make sense graphically???
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3.1 Derivatives. Derivative A derivative of a function is the instantaneous rate of change of the function at any point in its domain. We say this is.
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Page 13 Can you see the pattern? And indeed, (Bring the exponent to the front as a coefficient, and reduce the exponent by 1, and use that as the new exponent.)
AP Calculus Review First Semester Differentiation to the Edges of Integration Sections , 3.9, (7.7)
Differentiation – Product, Quotient and Chain Rules Department of Mathematics University of Leicester.
Lesson 7-1 Integration by Parts. Derived from the Product Rule in Differentiation D(uv) = v × du + u × dv ∫ d(uv) = ∫v du + ∫u dv uv = ∫v du + ∫u dv ∫u.
L’Hôpital’s Rule. L’Hopital’s Rule Analytically evaluate the following limit: By direct substitution you obtain the indeterminate form of type 0/0. 0/0,
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