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Calculus: Derivative Rules
Mastering derivative rules is fundamental to success in calculus. This flashcard deck provides 15 essential rules and common derivatives, perfect for quick review or solidifying your understanding. Focus on these core concepts to build a strong foundation for more advanced topics.
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What is the derivative of a constant function, d/dx (c)?
The derivative of any constant function is always zero.
State the Power Rule for differentiation, d/dx (x^n).
The Power Rule states that d/dx (x^n) = nx^(n-1), where n is any real number.
What is the Constant Multiple Rule for derivatives, d/dx (c * f(x))?
The Constant Multiple Rule states that d/dx (c * f(x)) = c * d/dx (f(x)). You can pull the constant out before differentiating.
State the Sum Rule for derivatives, d/dx (f(x) + g(x)).
The Sum Rule states that d/dx (f(x) + g(x)) = d/dx (f(x)) + d/dx (g(x)). The derivative of a sum is the sum of the derivatives.
State the Difference Rule for derivatives, d/dx (f(x) - g(x)).
The Difference Rule states that d/dx (f(x) - g(x)) = d/dx (f(x)) - d/dx (g(x)). The derivative of a difference is the difference of the derivatives.
What is the Product Rule for differentiation, d/dx (f(x) * g(x))?
The Product Rule states that d/dx (f(x) * g(x)) = f'(x)g(x) + f(x)g'(x). It's often remembered as 'first times derivative of second plus second times derivative of first'.
What is the Quotient Rule for differentiation, d/dx (f(x) / g(x))?
The Quotient Rule states that d/dx (f(x) / g(x)) = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2. A common mnemonic is 'low d-high minus high d-low over low-squared'.
State the Chain Rule for differentiation, d/dx (f(g(x))).
The Chain Rule states that d/dx (f(g(x))) = f'(g(x)) * g'(x). It's used for differentiating composite functions, multiplying the derivative of the outer function by the derivative of the inner function.
What is the derivative of sin(x)?
The derivative of sin(x) with respect to x is cos(x).
What is the derivative of cos(x)?
The derivative of cos(x) with respect to x is -sin(x).
What is the derivative of tan(x)?
The derivative of tan(x) with respect to x is sec^2(x).
What is the derivative of e^x?
The derivative of e^x with respect to x is e^x. It's its own derivative.
What is the derivative of ln(x)?
The derivative of ln(x) with respect to x is 1/x, for x > 0.
What is the derivative of a^x, where a is a constant base?
The derivative of a^x with respect to x is a^x * ln(a).
What is the derivative of sec(x)?
The derivative of sec(x) with respect to x is sec(x)tan(x).