12 Questions
What is the derivative of the function f(x) = sin(3x) using the chain rule?
cos(3x) * 3
What is the derivative of the function f(x) = cos(x^2) using the chain rule?
-2x * sin(x^2)
What is the derivative of the function f(x) = tan(x^2) using the chain rule?
2x * secant squared(x^2)
What is the derivative of the function f(x) = cosecant(x) using the derivative of cosecant?
-cosecant(x) * cotangent(x)
What is the derivative of the function f(x) = cotangent(x) using the derivative of cotangent?
-cosecant squared(x)
What is the derivative of the function f(x) = secant(x) using the derivative of secant?
secant(x) * tan(x)
What is the derivative of cosecant x?
- cosecant x cotangent x
What is the derivative of f(x) = x^3 + sin x + 4 cos x?
3x^2 + cos x - 4 sin x
What is the derivative of y = x^2 sin x?
2x sin x + x^2 cos x
What is the derivative of f(u) = sin(u), where u is a function of x?
cos(u) * u'
What is the derivative of y = (1 + sin x) / (x - tan x)?
(x - tan x)(cos x) - (1 + sin x)(1 - secant squared) / (x - tan x)^2
What is the chain rule formula for finding the derivative of a composite function f(g(x))?
f'(g(x)) * g'(x)
Study Notes
- The derivative of sine x is cosine x.
- The derivative of cosine x is - sine x.
- The derivative of tangent x is secant squared.
- The derivative of cosecant x is - cosecant x cotangent x.
- The derivative of secant x is secant x tan x.
- The derivative of cotangent x is - cosecant squared x.
- When there's a cosecant, cosine, or cotangent, there's usually a negative in front, which can help with memorization.
- The power rule is used to find the derivative of x^n, which is nx^(n-1).
- To find the derivative of a function, apply the derivative rules to each component of the function.
- The derivative of x^3 is 3x^2.
- The derivative of f(x) = x^3 + sin x + 4 cos x is 3x^2 + cos x - 4 sin x.
- The derivative of f(x) = 3 cosecant x + 9 tan x - 4 secant x is -3 cosecant x cotangent x + 9 secant squared - 4 secant x tan x.
- The derivative of y = x^2 sin x is 2x sin x + x^2 cos x, using the product rule.
- The derivative of y = x^3 cos x is 3x^2 cos x - x^3 sin x, using the product rule.
- The derivative of y = sin x / x^2 is (x cos x - 2 sin x) / x^3, using the quotient rule.
- The derivative of y = (1 + sin x) / (x - tan x) is (x - tan x)(cos x) - (1 + sin x)(1 - secant squared) / (x - tan x)^2, using the quotient rule.
- The chain rule is used to find the derivative of a composite function, f(g(x)), which is f'(g(x)) * g'(x).
- The chain rule can be written as f'(u) * u', where u = g(x).
- The derivative of sin(u) is cos(u) * u', where u is a function of x.
- The derivative of cos(u) is -sin(u) * u', where u is a function of x.
- The derivative of tan(u) is secant squared(u) * u', where u is a function of x.
- The derivative of cosecant(u) is -cosecant(u) * cotangent(u) * u', where u is a function of x.
- The derivative of secant(u) is secant(u) * tan(u) * u', where u is a function of x.
- The derivative of cotangent(u) is -cosecant squared(u) * u', where u is a function of x.
- The derivative of f(x) = sin(5x) is cos(5x) * 5, using the chain rule.- To find the derivative of a composite function, apply the chain rule by differentiating the outside function and then the inside function, starting from the outside and working inwards.
- The derivative of cosine is negative sine, and the derivative of sine is cosine.
- When differentiating a function, multiply the derivative of the outside function by the derivative of the inside function.
- The derivative of secant is secant tangent, and the derivative of cotangent is negative cosecant squared.
- To differentiate a function with a trigonometric function inside another, apply the chain rule and differentiate the outside function first, then the inside function.
- The power rule can be applied to rewrite a function before differentiating, such as rewriting sine squared as sine to the power of 2.
- When rewriting a function, keep the inside function the same and apply the power rule to the outside function.
- The derivative of x cubed is 3x squared, and the derivative of x squared is 2x.
- To prove that the derivative of secant is secant tangent, rewrite secant as 1 over cosine and use the power rule and chain rule to differentiate.
- To prove that the derivative of cotangent is negative cosecant squared, rewrite cotangent as cosine over sine and use the quotient rule to differentiate.
- The quotient rule formula is g(f') - f(g') divided by g squared, where f and g are functions.
- Sine squared plus cosine squared is equal to 1, and one over sine is cosecant.
- One over sine squared is cosecant squared, and negative one over sine squared is negative cosecant squared.
Test your knowledge of derivatives in calculus, covering trigonometric functions such as sine, cosine, and tangent, as well as the power rule for functions like x^n. Apply rules and formulas to find derivatives of composite functions and more.
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