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Differentiating powers, sin, cos and eˣ
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Differentiation gives the gradient function of a curve. Further Pure adds sin, cos and exponential functions to the powers of x.
Powers of x
If y = axn, then dydx = naxn − 1. This works for any rational n.
y = 2x2 = 2x−2 gives dydx = −4x−3
y = 6√x = 6x12 gives dydx = 3x−12
y = 2x2 = 2x−2 gives dydx = −4x−3
y = 6√x = 6x12 gives dydx = 3x−12
sin, cos and e
ddx(sin ax) = a cos ax
ddx(cos ax) = −a sin ax
ddx(eax) = aeax
Angles must be in radians. e = 2.718... is the special number whose exponential is its own derivative.
ddx(cos ax) = −a sin ax
ddx(eax) = aeax
Angles must be in radians. e = 2.718... is the special number whose exponential is its own derivative.
Gradient at a point
Differentiate, then substitute.
y = x3 − 4x + 1: dydx = 3x2 − 4. At x = 2 the gradient is 8.
y = x3 − 4x + 1: dydx = 3x2 − 4. At x = 2 the gradient is 8.
Differentiate y = 3 sin 2x + e4x − 5x2.
- 3 sin 2x → 3 × 2 cos 2x = 6 cos 2x
- e4x → 4e4x
- −5x2 → −10x
Answer: dydx = 6 cos 2x + 4e4x − 10x
axn → naxn − 1. sin ax → a cos ax. cos ax → −a sin ax. eax → aeax. Use radians.
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