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2. What is the derivative of 2^x? - Socratic

   socratic.org/questions/what-is-the-derivative-of-2-x

   This implies that 2x will be equivalent to. 2x = (eln2)x = ex⋅ln2. Your derivative now looks like this. d dx (ex⋅ln2) This is where the chain rule comes into play. You know that the derivative of a function y = f (u) can be written as. dy dx = dy du ⋅ du dx. In your case, y = ex⋅ln2, and u = x ⋅ ln2, so that your derivative becomes.

3. How do you find the derivative of 2^x? - Socratic

   socratic.org/questions/how-do-you-find-the-derivative-of-2-x

   Applying derivative, ⇒ 1 y dy dx = log2. ⇒ dy dx = 2xlog2. d/dx2^x=2^xln2 In general, the derivative of an exponential with some constant base is d/dxa^x=a^xlna. A proof of this will be shown. So, d/dx2^x=2^xln2 Proof: Rewrite a^x as e^ln (a^x)=e^ (xlna).

4. How do find the derivative of # y = cos^2(x)#? - Socratic

   socratic.org/questions/how-do-find-the-derivative-of-y-cos-2-x

   Calculus Differentiating Trigonometric Functions Derivative Rules for y=cos(x) and y=tan(x)

5. What is the derivative of #tanh(x)#? - Socratic

   socratic.org/questions/what-is-the-derivative-of-tanh-x

   The derivative is: 1 −tanh2(x) Hyperbolic functions work in the same way as the "normal" trigonometric "cousins" but instead of referring to a unit circle (for sin,cos and tan) they refer to a set of hyperbolae. (Picture source: Physicsforums.com) You can write: tanh(x) = ex −e−x ex +e−x. It is now possible to derive using the rule of ...

6. What is the derivative of #csc^2(x)#? - Socratic

   socratic.org/questions/what-is-the-derivative-of-csc-2-x

   cosx sinx = cotx ⇒ − 2cosx sin3x = − 2cotx sin2x. 1 sin2x = csc2x ⇒ −2cotxcsc2x. d dx [csc2(x)] = − 2cotxcsc2x. Answer link. iOS. Android.

7. How do you differentiate #2^(2x)#? - Socratic

   socratic.org/questions/how-do-you-differentiate-2-2x

   f'(x) = 2^(1+2x)ln 2 Given: f(x) = 2^(2x) Derivative rule: (a^u)' = u' a^u ln (a) Let a = 2; " "u = 2x; " "u' = 2 f'(x) = 2* 2^(2x) ln 2 f'(x) = 2^1 * 2^(2x) ln 2 Exponents of the same base are added: f'(x) = 2^(1+2x) ln 2

8. What is the derivative of #sin^2(x)#? - Socratic

   socratic.org/questions/what-is-the-derivative-of-sin-2-x

   The outer function of u^2 squares the inner function of u=sin (x). Don't let the u confuse you, it's just to show you how one function is a composite of the other. Once you understand this, you can derive. So, mathematically, the chain rule is: The derivative of a composite function F (x) is: F' (x)=f' (g (x)) (g' (x)) Or, in words: the ...

9. Limit Definition of Derivative - Calculus - Socratic

   socratic.org/calculus/derivatives/limit-definition-of-derivative

   Remember that the limit definition of the derivative goes like this: f '(x) = lim h→0 f (x + h) − f (x) h. So, for the posted function, we have. f '(x) = lim h→0 m(x + h) + b − [mx +b] h. By multiplying out the numerator, = lim h→0 mx + mh + b − mx −b h. By cancelling out mx 's and b 's, = lim h→0 mh h. By cancellng out h 's,

10. What is the derivative of tan^2 x? - Socratic

    socratic.org/questions/what-is-the-derivative-of-tan-2-x

    Explanation: note tan2x = (tanx)2. differentiate using the chain rule. given y = f (g(x)) then. dy dx = f '(g(x)) × g'(x) ← chain rule. y = (tanx)2. ⇒ dy dx = 2tanx × d dx (tanx) ⇒ dy dx = 2tanxsec2x. Answer link.

11. How do you find the derivative of #f(x)=sqrt(a^2+x^2)#? -...

    socratic.org/questions/how-do-you-find-the-derivative-of-f-x-sqrt-a-2-x-2

    How do you find the derivative of #f(x)=sqrt(a^2+x^2)#? Calculus Basic Differentiation Rules Chain Rule. 1 Answer