9–61. Evaluate and simplify y'.

y = 10^s...

Question

9–61. Evaluate and simplify y'.

y = 10^sin x+sin¹⁰x

Accepted Answer

Hello, in this video, we are going to be calculating the derivative of the given function. The function given to us in this problem is G of T is equal to 2 race of the power of tangent of T, plus tangent squared of T. So, we will go ahead and take the derivative with respect to T, but how do we start this process? Well, in order to take the derivative of the GFT function, we will have to take the individual derivatives of each of the terms in the function. Let's go ahead and start by taking the derivative of to race the power of tangent of T. Now, for the first term, notice that the variable T is in the form of an exponential value. In order to take the derivative of the first term, we will need to use the exponential rule. Of the derivative Now, in order to take the derivative of to raise the power of tangent of T, the exponential rule states that we first take the original term to race the power of tangent of T. Then we want to go ahead and multiply this value. By the natural logarithm of the base of the exponential value, which is 2. So we will multiply this by the natural logarithm of 2. Then we will also multiply the value by the derivative of the exponential function, which in this case is going to be tangent of T. So there's a little bit of simplification to do here, but the only simplification we need to do for the first term is take the derivative of tangent of T with respect to T. The derivative of tangent will be seekin squared. So the derivative of the first term of GFT will simplify to be to raise the power of tangent of T, multiplied by the natural logarithm of 2, multiplied by sequent squared of T. Now we need to take the derivative of tangent squared of T. Now tangent squared of t. Can be rewritten. As tangent of T raise to the power of 2. In order to take the derivative of this term, we are going to use the chain rule. So, the chain rule states that we first need to take the derivative of the outermost function. The outermost function with respect to T is going to be the exponential value of 2. So first, we will take the derivative using the power rule of the outermost function. That is going to give us 2 multiplied by the inside function, which is tangent of T, raise to the power of 2 minus 1, which will be 1. Then we need to multiply this function by the derivative of the innermost function. And again the derivative of tangent with respect to t will be seek and squared. So In order to simplify this, the only thing we need to realize is that tangent rates to the power of one is just tangent. So the derivative of the second term is going to be 2 multiplied by tangent of T. Multiplied by sequent squared of T. Now that we have the derivatives of the individual terms, let's go ahead and plug them back into the sum. So that gives us G T. is equal to 2 raised to the power of tangent of T multiplied by the natural logarithm of 2 multiplied by sequin squared of T. Plus the derivative of the second term, which is 2 multiplied by tangent oft multiplied by sequent squared of t. There is one more simplification we can do to the derivative. Notice that each of the terms have a multiple of sequin squared of T. So if we factor out a skin squared of T from the derivative, we can simplify the derivative to be skin squad of T multiplied by the quantity, to raise the power of tangent of T multiplied by the natural logarithm of 2, plus 2 multiplied by tangent of T. And this is going to be the simplest form of the derivative of the GFT function. And with that being said, the solution to this problem is going to be C. So I hope this video helps you in understanding how to approach this type of derivative, and I will go ahead and see you all in the next video.

9–61. Evaluate and simplify y'.

y = 10^sin x+sin¹⁰x

Key Concepts

Differentiation

Chain Rule

Power Rule

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