Derivatives of Composite Functions

Derivatives of Composite Functions If y = f(g(x)) then y' = f'(g(x))g'(x) y = (1 - 2x^4)^7 y' = 7(1 - 2x^4)^6(-8x^3) y' = -56x^3(1 - 2x^4)^6 Find the derivative y' of each function. 1) y = 4x^4 + 12x^2 + 9 y' = 16x^3 + 24x 2) y = x^3 + 6x^2 + 12x + 8 y' = 3x^2 + 12x + 12 3) y = 1 - 10x^3 + 25x^6 y' = -30x^2 + 150x^5 4) y = (2x^2 + 3)^2 y' = 2(2x^2 + 3)(4x) 5) y = (x + 2)^3 y' = 3(x + 2)^2(1) 6) y = (1 - 5x^3)^2 y' = 2(1 - 5x^3)(-15x^2) 7) y = 3(x^2 + 2x)^2 y' = 3 * 2(x^2 + 2x)(2x + 2) 8) y = (1 + √x)^2 = (1 + x^(1/2))^2 y' = 2(1 + x^(1/2))(1/2x^(-1/2)) 9) y = (1 - 1/x)^2 = (1 - x^-1)^2 y' = 2(1 - x^-1)(-x^-2) 10) y = (2x^2 + 3)^5 y' = 5(2x^2 + 3)^4(4x) 11) y = √(x^2 - 4) = (x^2 - 4)^(1/2) y' = (1/2)(x^2 - 4)^(-1/2)(2x) 12) y = -5√(2 - x) = -5(2 - x)^(1/2) y' = -5 * (1/2)(2 - x)^(-1/2)(-1) 13) y = -1/(x^2 - 4) = -(x^2 - 4)^-1 y' = -(-1)(x^2 - 4)^-2(2x) 14) y = -1/√(x^2 - 4) = -(x^2 - 4)^(-1/2) y' = -(-1/2)(x^2 - 4)^(-3/2)(2x) 15) y = -5√(2 - x) = -5(2 - x)^(1/2) y' = -5 * (1/2)(2 - x)^(-1/2)(-1) Derivatives E. y' = 8x(2x^2 + 3) O. y' = -30x^2(1 - 5x^3) R. y' = 12x(x + 1)(x + 2) S. y' = 3(x + 2)^2 C. y' = 40x(2x^2 + 3)^4 L. y' = x / (x^2 - 4)^(3/2) M. y' = 2x / (x^2 - 4)^2 P. y' = x / √(x^2 - 4) U. y' = 5 / 3(2 - x)^(3/2) V. y' = (1 + √x) / √x W. y' = 2/x^2 - 2/x^3 Y. y' = 5 / 2√(2 - x) COMPOSE YOURSELVES Find the derivative of each function and match with the result. 1) y = 2sin(x) 2) y = 3cos(x) 3) y = 2tan(x) + 1 4) y = sin(x - π) 5) y = tan(x - π) + 4 6) y = -2cos(x/2) + 4 7) y = 2sin(x/2 - π/2) - 1 Derivatives A. y' = 2sin(x) E. y' = 2sec²(x) G. y' = sin(x/2) I. y' = 2cos(x) M. y' = 2cot(2x) N. y' = cos(x - π) O. y' = sin(x - π) P. y' = -3sin(x) S. y' = sec²(x - π) T. y' = cos(x/2) U. y' = cos(x/2 - π/2) #2 Find the derivative of each function and match with the result. 1) y = sin(2x) 2) y = cos(x²) 3) y = 2tan(2x) 4) y = xsin(x) 5) y = 1/4 tan(2x²) 6) y = -cos(√x) 7) y = 1/x tan(x) 8) y = cos(2π - x) Derivatives A. y' = 2sin(2x) E. y' = 4xsec²(2x²) G. y' = -2xsin(x²) I. y' = 2cos(2x) M. y' = 2tan(2x) N. y' = sin(2π - x) O. y' = xcos(x) + sin(x) P. y' = 1/2√x sin(√x) S. y' = 1/x sec²(x) - 1/x² tan(x) T. y' = cos(x/2) WHAT IS THE TITLE OF THIS PICTURE? SIGN Derivatives of the Six Trigonometric Functions: y = sin(x) -> y' = cos(x) y = cos(x) -> y' = -sin(x) y = tan(x) -> y' = sec²(x) y = cot(x) -> y' = -csc²(x) y = sec(x) -> y' = sec(x)tan(x) y = csc(x) -> y' = -csc(x)cot(x) Find each derivative: 1) y = sin(3x) y' = 3cos(3x) 2) y = csc(3x) y' = -3csc(3x)cot(3x) 3) y = 3tan(x) y' = 3sec²(x) 4) y = -cos(2x) y' = 2sin(2x) 5) y = cos(x + π/3) y' = -sin(x + π/3) 6) y = cos(π/3 - x) y' = sin(π/3 - x) 7) y = 1/3sec(3x) y' = sec(3x)tan(3x) 8) y = sin(1/3x) y' = 1/3cos(1/3x) 9) y = 1/2cos(x²) y' = -xsin(x²) 10) y = 1/2cot(x²) y' = -csc²(x²)/x 11) y = 1/3tan(x³) y' = x²sec²(x³) 12) y = cot(π) y' = 0 13) y = sin²(x) - (cos(x))² y' = 2sin(x)cos(x) + 2sin(x)cos(x) y' = 4sin(x)cos(x) 14) y = 1/2cos(1/x²) y' = (1/2)(-sin(1/x²))(-2/x³) y' = sin(1/x²)/x³ 15) y = sin(sec(x)) y' = cos(sec(x))sec(x)tan(x) Derivatives: A. y' = 0 B. y' = -1/3cos(1/3x) C. y' = 3cos(3x) E. y' = -3csc(3x)cot(3x) G. y' = sin(x/3 - x) I. y' = 2cos(2x) K. y' = -xcsc²(x²) L. y' = 4xsin(2x²) M. y' = 2tan(2x) N. y' = sin(2x) O. y' = xcos(x) + sin(x) P. y' = 1/2√x sin(√x) S. y' = sec(3x)tan(3x) T. y' = cos(x/2) U. y' = sec(x)[sec²(x) + tan²(x)] IT'S CALLED A SECANT